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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 9 /Psi put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 16 /dotlessi put dup 18 /grave put dup 19 /acute put dup 23 /ring put dup 24 /cedilla put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 105 /i put dup 107 /k put dup 110 /n put dup 112 /p put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 54 /lessorequalslant put dup 62 /greaterorequalslant put readonly def /FontBBox{8 -463 1331 1003}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: Yhcmex %!PS-AdobeFont-1.0: Yhcmex 001.000 %%Title: Yhcmex %%CreationDate: Wed Aug 6 21:06:51 2003 %%Creator: Haruhiko Okumura %%DocumentSuppliedResources: font Yhcmex % Generated by PfaEdit 1.0 (http://pfaedit.sf.net/) %%EndComments FontDirectory/Yhcmex known{/Yhcmex findfont dup/UniqueID known{dup /UniqueID get 4731634 eq exch/FontType get 1 eq and}{pop false}ifelse {save true}{false}ifelse}{false}ifelse 11 dict begin /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0 ]readonly def /FontName /Yhcmex def /FontBBox [-17 -6569 4798 931 ]readonly def /UniqueID 4731634 def 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cleartomark {restore}if %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 23 /ring put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 97 /a put dup 105 /i put dup 109 /m put dup 110 /n put dup 120 /x put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 23 /nu put dup 25 /pi put dup 27 /sigma put dup 39 /phi1 put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 65 /A put dup 69 /E put dup 72 /H put dup 73 /I put dup 80 /P put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 105 /i put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 117 /u put dup 118 /v put dup 120 /x put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 72 /H put dup 94 /logicaland put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 14 /openbullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 71 /G put dup 72 /H put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 91 /union put dup 92 /intersection put dup 94 /logicaland put dup 95 /logicalor put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 16 /dotlessi put dup 18 /grave put dup 19 /acute put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSL12 %!PS-AdobeFont-1.1: CMSL12 1.0 %%CreationDate: 1991 Aug 20 16:40:41 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 18 /grave put dup 19 /acute put dup 39 /quoteright put dup 44 /comma put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 127 /dieresis put readonly def /FontBBox{-56 -251 1102 750}readonly def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 18 /grave put dup 19 /acute put dup 39 /quoteright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 89 /Y put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 123 /endash put readonly def /FontBBox{-36 -251 1103 750}readonly def currentdict end currentfile eexec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b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.) 89 b(35)578 1977 y @beginspecial @setspecial @endspecial(4.2)99 b(D)m(\023)-46 b(ecomp)s(osition)33 b(d'un)h(nom)m(bre)f(en)g(pro)s (duit)g(de)g(facteurs)g(premiers)p @beginspecial @setspecial @endspecial 81 w(.)89 b(36)802 2097 y @beginspecial @setspecial @endspecial(4.2.1)112 b(Existence)35 b(et)e(unicit)m(\023)-46 b(e)33 b(de)g(la)g(d)m(\023)-46 b(ecomp)s(osition)p @beginspecial @setspecial @endspecial 84 w(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)89 b(36)802 2218 y @beginspecial @setspecial @endspecial(4.2.2)112 b(Application)33 b(au)f(calcul)i(du)f(PGCD)f(et)h(du)g(PPCM)p @beginspecial @setspecial @endspecial 86 w(.)50 b(.)g(.)g(.)g(.)89 b(37)578 2338 y @beginspecial @setspecial @endspecial(4.3)99 b(Exercices)p @beginspecial @setspecial @endspecial 32 w(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)89 b(39)432 2556 y @beginspecial @setspecial @endspecial Fx(5)h Fu(Z)p Fs(=n)p Fu(Z)p @beginspecial @setspecial @endspecial 2740 w Fx(41)578 2676 y @beginspecial @setspecial @endspecial Fy(5.1)99 b(Construction)34 b(de)f(l'anneau)h Fu(Z)p Fs(=n)p Fu(Z)p @beginspecial @setspecial @endspecial 78 w Fy(.)49 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g (.)g(.)89 b(41)578 2797 y @beginspecial @setspecial @endspecial(5.2)99 b(In)m(v)m(ersibles)36 b(de)d Fu(Z)p Fs(=n)p Fu(Z)p @beginspecial @setspecial @endspecial 38 w Fy(.)50 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)89 b(41)802 2917 y @beginspecial @setspecial @endspecial(5.2.1)112 b(Caract)m(\023)-46 b(erisation)33 b(des)g(in)m(v)m(ersibles)p @beginspecial @setspecial @endspecial 56 w(.)50 b(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)89 b(41)802 3038 y @beginspecial @setspecial @endspecial(5.2.2)112 b(Group)s(e)32 b(des)f(\023)-46 b(el)m(\023)g(emen)m(ts)34 b(in)m(v)m(ersibles)i({)c(indicatrice)i (d'Euler)p @beginspecial @setspecial @endspecial 87 w(.)89 b(42)578 3158 y @beginspecial @setspecial @endspecial(5.3)99 b(Th)m(\023)-46 b(eor)m(\022)g(eme)35 b(c)m(hinois)p @beginspecial @setspecial @endspecial 63 w(.)50 b(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.) g(.)89 b(43)578 3278 y @beginspecial @setspecial @endspecial(5.4)99 b(Calcul)34 b(de)f(l'indicatrice)h(d'Euler)p @beginspecial @setspecial @endspecial 73 w(.)50 b(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)89 b(44)578 3399 y @beginspecial @setspecial @endspecial(5.5)99 b(Co)s(dage)33 b(RSA)p @beginspecial @setspecial @endspecial 29 w(.)49 b(.)h(.)g(.)g(.)g(.)g (.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.) h(.)g(.)g(.)g(.)89 b(45)578 3519 y @beginspecial @setspecial @endspecial(5.6)99 b(Exercices)p @beginspecial @setspecial @endspecial 32 w(.)50 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)89 b(47)p eop end %%Page: 1 5 TeXDict begin 1 4 bop 83 291 a @beginspecial @setspecial @endspecial 165 x @beginspecial @setspecial @endspecial 763 x Fv(Chapitre)77 b(1)83 1634 y(Construction)h(de)g Fr(N)83 1967 y @beginspecial @setspecial @endspecial 182 x Fq(1.1)161 b(Lois)53 b(de)g(comp)t(osition)f(in)l(terne)83 2368 y Fx(D)n(\023)-54 b(e\014nition:)40 b Fy(Soit)33 b Fs(X)42 b Fy(un)34 b(ensem)m(ble.)i(On)e(app)s(elle)h(loi)e(de)i (comp)s(osition)f(in)m(terne)h(sur)83 2489 y Fs(X)40 b Fy(toute)33 b(application)g Fs(?)f Fy(de)h Fs(X)d Fp(\002)23 b Fs(X)40 b Fy(dans)33 b Fs(X)41 b Fy(:)1103 2681 y Fs(X)30 b Fp(\002)23 b Fs(X)90 b Fp(!)83 b Fs(X)1176 2826 y Fy(\()p Fs(x;)17 b(y)t Fy(\))82 b Fp(7!)h Fs(x)23 b(?)f(y)30 b Fy(:=)e Fs(?)p Fy(\()p Fs(x;)17 b(y)t Fy(\))83 3018 y(Ici,)33 b(on)g(c)m(hoisira)g(d'utiliser)h(la)f Fs(x)22 b(?)g(y)36 b Fy(plut^)-49 b(ot)32 b(que)i Fs(?)p Fy(\()p Fs(x;)17 b(y)t Fy(\).)32 b(V)-8 b(oici)32 b(p)s(ourquoi)h(:)83 3139 y Fx(D)n(\023)-54 b(e\014nition:)39 b Fy(Soit)33 b Fs(X)41 b Fy(un)34 b(ensem)m(ble)i(et)d Fs(?)g Fy(une)h(loi)f(de)h (comp)s(osition)g(in)m(terne)g(sur)g Fs(X)8 b Fy(.)83 3259 y(On)33 b(dit)f(que)i Fs(?)e Fy(est)h(asso)s(ciativ)m(e)i(si)e(on) f(a)666 3451 y Fp(8)p Fy(\()p Fs(x;)17 b(y)t(;)g(z)t Fy(\))28 b Fp(2)g Fs(X)i Fp(\002)22 b Fs(X)30 b Fp(\002)23 b Fs(X)105 b Fy(\()p Fs(x)23 b(?)f(y)t Fy(\))f Fs(?)h(z)32 b Fy(=)c Fs(x)22 b(?)g Fy(\()p Fs(y)j(?)d(z)t Fy(\))83 3643 y(Ainsi,)30 b(p)s(our)e(une)h(loi)g(de)g(comp)s(osition)g(in)m (terne)h(asso)s(ciativ)m(e,)g(on)f(p)s(eut)d(\023)-46 b(ecrire)29 b Fs(x)14 b(?)g(y)j(?)d(z)83 3764 y Fy(sans)33 b(que)h(cel\022)-49 b(a)33 b(soit)g(source)h(d'am)m(bigu)-11 b(\177)-38 b(\020t)m(\023)-46 b(e.)83 3884 y Fx(Exemples:)36 b Fy(+)30 b(et)g Fp(\002)g Fy(son)m(t)h(des)g(lois)g(de)f(comp)s (osition)h(in)m(ternes)h(sur)e Fu(R)p Fy(.)h Fs(=)e Fy(est)i(une)g(loi) 83 4004 y(de)i(comp)s(osition)g(in)m(terne)h(sur)f(]0)p Fs(;)17 b Fy(+)p Fp(1)p Fy([.)229 4125 y(De)32 b(plus,)i(+)e(et)g Fp(\002)h Fy(son)m(t)g(des)g(lois)f(de)h(comp)s(osition)g(in)m(ternes)h (asso)s(ciativ)m(es)g(sur)f Fu(R)p Fy(.)83 4245 y(Cep)s(endan)m(t,)28 b Fs(=)f Fy(n'est)g(pas)g(une)h(loi)e(de)h(comp)s(osition)g(in)m(terne) h(asso)s(ciativ)m(e)h(sur)e(]0)p Fs(;)17 b Fy(+)p Fp(1)p Fy([)83 4365 y(car)33 b(1)p Fs(=)p Fy(\(1)p Fs(=)p Fy(2\))26 b(=)i(2)k(n'est)h(pas)e(\023)-46 b(egal)32 b(\022)-49 b(a)32 b(\(1)p Fs(=)p Fy(1\))p Fs(=)p Fy(2)27 b(=)g(1)p Fs(=)p Fy(2.)83 4486 y Fx(D)n(\023)-54 b(e\014nition:)46 b Fy(On)40 b(dit)g(qu'un)e(\023)-46 b(el)m(\023)g(emen)m(t)41 b Fs(e)e Fy(est)i(neutre)f(p)s(our)f(la)h(loi)f(de)h(comp)s(osition)83 4606 y(in)m(terne)34 b Fs(?)e Fy(sur)h Fs(X)41 b Fy(si)1126 4727 y Fp(8)p Fs(x)29 b Fp(2)f Fs(X)105 b(e)23 b(?)e(x)29 b Fy(=)e Fs(x)c(?)f(e)28 b Fy(=)f Fs(x:)229 4889 y Fy(Il)h(y)f(a)g (toujours)g(au)g(plus)g(un)e(\023)-46 b(el)m(\023)g(emen)m(t)29 b(neutre,)f(car)f(si)g Fs(e)g Fy(et)g Fs(e)2470 4853 y Fo(0)2521 4889 y Fy(son)m(t)g(deux)f(\023)-46 b(el)m(\023)g(emen)m (ts)83 5009 y(neutre)34 b Fs(e)27 b Fy(=)h Fs(e)22 b(?)g(e)746 4973 y Fo(0)797 5009 y Fy(=)28 b Fs(e)946 4973 y Fo(0)969 5009 y Fy(.)83 5130 y Fx(D)n(\023)-54 b(e\014nition:)44 b Fy(Le)38 b(couple)g(\()p Fs(X)r(;)17 b(?)p Fy(\))37 b(form)m(\023)-46 b(e)38 b(par)f(un)h(ensem)m(ble)i Fs(X)46 b Fy(et)37 b(une)h(loi)g(de)g(com-)83 5250 y(p)s(osition)31 b(in)m(terne)g(asso)s(ciativ)m(esur)i Fs(X)38 b Fy(p)s(oss)m(\023)-46 b(edan)m(t)31 b(un)d(\023)-46 b(el)m(\023)g(emen)m(t)32 b(neutre)f(est)h(app)s(el)m(\023)-46 b(e)30 b(un)83 5370 y(mono)-11 b(\177)-38 b(\020de.)1678 5620 y(1)p eop end %%Page: 2 6 TeXDict begin 2 5 bop 432 291 a @beginspecial @setspecial @endspecial Fy(2)1399 b Ft(CHAPITRE)34 b(1.)65 b(CONSTR)m(UCTION)35 b(DE)e Fu(N)432 555 y Fx(Exemples:)52 b Fy(Si)44 b Fp(G)50 b Fy(est)45 b(une)g(famille)g(d'applications)g(d'un)g(ensem)m(ble)i Fs(E)j Fy(dans)45 b(lui)432 676 y(m)m(^)-46 b(eme)34 b(telle)f(que)578 796 y({)49 b(Id)765 811 y Fn(E)852 796 y Fp(2)28 b(G)6 b Fy(.)578 917 y({)49 b Fp(8)p Fy(\()p Fs(f)5 b(;)17 b(g)t Fy(\))27 b Fp(2)h(G)g(\002)23 b(G)104 b Fs(f)32 b Fp(\016)22 b Fs(g)31 b Fp(2)d(G)6 b Fy(,)432 1037 y(alors)32 b(\()p Fp(G)6 b Fs(;)17 b Fp(\016)p Fy(\))32 b(est)i(un)f(mono)-11 b(\177)-38 b(\020de)578 1158 y(P)m(ar)29 b(exemple,)h(l'ensem)m(ble)i(des)d(applications)h(a\016nes)f(sur)g Fu(R)f Fy(forme)h(un)g(mono)-11 b(\177)-38 b(\020de.)432 1279 y Fx(D)n(\023)-54 b(e\014nition:)54 b Fy(On)47 b(dit)g(qu'un)e (\023)-46 b(el)m(\023)g(emen)m(t)48 b Fs(x)f Fy(d'un)h(mono)-11 b(\177)-38 b(\020de)47 b(\()p Fs(X)r(;)17 b(?)p Fy(\))46 b(don)m(t)i(l')m(\023)-46 b(el)m(\023)g(emen)m(t)432 1399 y(neutre)33 b(est)g(not)m(\023)-46 b(e)33 b Fs(e)g Fy(est)g(in)m(v)m(ersible)j(s'il)d(existe)h(un)f Fs(y)e Fp(2)d Fs(X)40 b Fy(tel)33 b(que)1684 1621 y Fs(x)22 b(?)g(y)31 b Fy(=)c Fs(y)f(?)c(x)28 b Fy(=)f Fs(e:)432 1842 y Fy(S'il)22 b(existe,)i(un)e(tel)d(\023)-46 b(el)m(\023)g(emen)m (t)24 b(est)e(n)m(\023)-46 b(ecessairemen)m(t)25 b(unique)f(:)d(en)i (e\013et)f(si)g Fs(x?y)32 b Fy(=)27 b Fs(y)t(?x)h Fy(=)f Fs(e)432 1962 y Fy(et)33 b Fs(x)22 b(?)g(y)746 1926 y Fo(0)796 1962 y Fy(=)28 b Fs(y)952 1926 y Fo(0)996 1962 y Fs(?)22 b(x)28 b Fy(=)g Fs(e)p Fy(,)33 b(alors)1002 2183 y Fs(y)1054 2142 y Fo(0)1104 2183 y Fy(=)28 b Fs(y)1260 2142 y Fo(0)1304 2183 y Fs(?)22 b(e)28 b Fy(=)g Fs(y)1604 2142 y Fo(0)1648 2183 y Fs(?)22 b Fy(\()p Fs(x)h(?)f(y)t Fy(\))k(=)i(\()p Fs(y)2216 2142 y Fo(0)2261 2183 y Fs(?)21 b(x)p Fy(\))i Fs(?)f(y)31 b Fy(=)c Fs(e)c(?)e(y)31 b Fy(=)d Fs(y)t(:)578 2405 y Fy(Usuellemen)m(t,)h(on)d(note)h Fs(x)1547 2369 y Fo(\000)1606 2405 y Fy(1)f(l'in)m(v)m(erse)j(de)e Fs(x)f Fy(lorsqu'il)i(existe.)g(\(On)e(trouv)m(era)h(plus)432 2525 y(raremen)m(t)32 b Fs(x)900 2489 y Fn(?)p Fo(\000)p Fm(1)1030 2525 y Fy(\))f(:)g(cette)g(notation)g(n'est)h(emplo)m(y)m (\023)-46 b(ee)33 b(que)f(si)f(plusieurs)i(structures)g(de)432 2646 y(group)s(e)22 b(pson)m(t)i(mises)g(sur)f(l'ensem)m(ble)i(consid)m (\023)-46 b(er)m(\023)g(e)24 b(et)f(qu'une)h(am)m(biguit)m(\023)-46 b(e)24 b(est)f(p)s(ossible.\))432 2766 y(l)432 2918 y @beginspecial @setspecial @endspecial 183 x Fq(1.2)160 b(Group)t(es,)53 b(sous-group)t(es)432 3321 y Fx(D)n(\023)-54 b(e\014nition:)47 b Fy(Un)41 b(mono)-11 b(\177)-38 b(\020de)41 b(\()p Fs(G;)17 b(?)p Fy(\))39 b(don)m(t)i(tous)g(les)e(\023)-46 b(el)m(\023)g(emen)m(ts)42 b(son)m(t)f(in)m(v)m(ersibles)j(est)432 3441 y(app)s(el)m(\023)-46 b(e)33 b(un)g(group)s(e.)432 3562 y Fx(Exemples:)d Fy(Si)c Fp(G)31 b Fy(est)26 b(une)g(famille)g(de) g(bijections)g(d'un)g(ensem)m(ble)i Fs(E)k Fy(dans)26 b(lui)f(m)m(^)-46 b(eme)432 3682 y(telle)33 b(que)578 3803 y({)49 b(Id)765 3818 y Fn(E)852 3803 y Fp(2)28 b(G)6 b Fy(.)578 3924 y({)49 b Fp(8)p Fy(\()p Fs(f)5 b(;)17 b(g)t Fy(\))27 b Fp(2)h(G)g(\002)23 b(G)104 b Fs(f)32 b Fp(\016)22 b Fs(g)31 b Fp(2)d(G)6 b Fy(,)432 4044 y(alors)32 b(\()p Fp(G)6 b Fs(;)17 b Fp(\016)p Fy(\))32 b(est)i(un)f(group)s(e.) 578 4165 y(Lorsque)j Fs(X)42 b Fy(est)35 b(un)g(ensem)m(ble)i(\014ni,)e (on)f(note)h Fp(S)7 b Fy(\()p Fs(X)h Fy(\))35 b(le)g(group)s(e)f(form)m (\023)-46 b(e)35 b(par)f(l'en-)432 4286 y(sem)m(ble)g(des)g(bijections) g(de)f Fs(X)40 b Fy(dans)33 b(lui-m)m(^)-46 b(eme)34 b(m)m(uni)f(de)h(la)e(comp)s(osition)578 4406 y(T)-8 b(raditionnellemen)m(t,)34 b(on)d(app)s(elle)h(\\p)s(erm)m(utations")g (les)e(\023)-46 b(el)m(\023)g(emen)m(ts)33 b(de)f Fp(S)7 b Fy(\()p Fs(X)h Fy(\),)32 b(et,)432 4527 y(fort)g(logiquemen)m(t,)i (on)f(app)s(elle)g(ce)g(group)s(e)g(le)g(group)s(e)f(des)i(p)s(erm)m (utations.)432 4647 y Fx(D)n(\023)-54 b(e\014nition:)37 b Fy(Soit)31 b(\()p Fs(G;)17 b(?)p Fy(\))31 b(un)h(group)s(e,)f Fs(H)39 b Fy(une)32 b(partie)g(de)g Fs(G)p Fy(.)f(On)h(dit)g(que)g Fs(H)39 b Fy(est)32 b(un)432 4768 y(sous-group)s(e)h(de)g Fs(G)f Fy(si)h(\()p Fs(H)r(;)17 b(?)p Fy(\))32 b(un)h(group)s(e.)578 4889 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Fs(D)g Fy(=)d Fp(f)p Fs(p)f Fp(2)i Fu(N)p Fy(;)17 b Fp(8)p Fs(n)33 b Fp(2)g Fu(N)98 b Fs(n)24 b Fy(+)h Fs(p)32 b Fy(=)h Fs(p)25 b Fy(+)f Fs(n)p Fy(.)36 b(Comme)g(0)g(est)p eop end %%Page: 5 9 TeXDict begin 5 8 bop 83 291 a @beginspecial @setspecial @endspecial Ft(1.4.)65 b Fu(N)p Ft(,)32 b(CE)i(MONO)912 265 y(\177)915 291 y(IDE)2181 b Fy(5)83 555 y(neutre)34 b(p)s(our)e(+,)g(on)h(sait)g(que)g(0)28 b Fp(2)g Fs(B)5 b Fy(.)32 b(Mon)m(trons)i(que)f Fs(p)28 b Fp(2)g Fs(D)j Fp(\000)-17 b(!)28 b Fs(s)p Fy(\()p Fs(p)p Fy(\))f Fp(2)h Fs(D)s Fy(.)304 786 y Fs(n)23 b Fy(+)f Fs(s)p Fy(\()p Fs(p)p Fy(\))83 b(=)f Fs(s)p Fy(\()p Fs(n)23 b Fy(+)f Fs(p)p Fy(\))206 b Fz(d)n(\023)-47 b(e\014nition)33 b(de)h(l'addition) 737 907 y Fy(=)82 b Fs(s)p Fy(\()p Fs(p)22 b Fy(+)g Fs(n)p Fy(\))207 b Fz(hyp)-5 b(oth)n(\022)-47 b(ese)33 b(de)i(r)n(\023)-47 b(ecurr)-5 b(enc)g(e)737 1027 y Fy(=)82 b(\()p Fs(p)23 b Fy(+)f Fs(n)p Fy(\))g(+)g(1)83 b Fz(p)-5 b(ar)34 b(d)n(\023)-47 b(e\014nition)737 1147 y Fy(=)82 b Fs(p)23 b Fy(+)f(\()p Fs(n)g Fy(+)g(1\))83 b Fz(asso)-5 b(ciativit)n(\023)-47 b(e)737 1268 y Fy(=)82 b Fs(p)23 b Fy(+)f(\(1)f(+)h Fs(n)p Fy(\))84 b Fz(d'apr)n(\022)-47 b(es)33 b(la)h(pr)-5 b(opri)n(\023)-47 b(et)n(\023)g(e)33 b(montr)n(\023)-47 b(ee)33 b(au)i(dessus)737 1388 y Fy(=)82 b(\()p Fs(p)23 b Fy(+)f(1\))f(+)h Fs(n)84 b Fz(hyp)-5 b(oth)n(\022)-47 b(ese)33 b(de)i(r)n(\023)-47 b(ecurr)-5 b(enc)g(e)737 1508 y Fy(=)82 b Fs(s)p Fy(\()p Fs(p)p Fy(\))22 b(+)g Fs(n)207 b Fz(p)-5 b(ar)34 b(d)n(\023)-47 b(e\014nition)83 1708 y @beginspecial @setspecial @endspecial 96 x Fw(1.4.3)136 b(Relation)47 b(d'ordre)e(sur)g Fj(N)229 1998 y Fy(On)33 b(d)m(\023)-46 b(e\014nit)33 b(une)h(relation)e Fi(6)h Fy(binaire)h(sur)f Fs(N)43 b Fy(par)764 2232 y Fp(8)p Fs(a;)17 b(b)29 b Fp(2)f Fu(N)1150 2191 y Fm(2)1287 2232 y Fs(a)g Fi(6)g Fs(b)55 b Fp(\()-17 b(\))55 b(9)p Fs(n)29 b Fp(2)f Fu(N)97 b Fs(a)23 b Fy(+)f Fs(n)28 b Fy(=)f Fs(b:)83 2466 y Fi(6)33 b Fy(est)g(bien)h(une)f(relation)g (d'ordre)g(car)f(on)h(a)229 2591 y({)49 b(R)m(\023)-46 b(e\015exivit)m(\023)g(e)35 b(:)d(p)s(our)h(tout)f(en)m(tier)i(naturel) f Fs(a)p Fy(,)f Fs(a)c Fi(6)g Fs(a)33 b Fy(car)g Fs(a)28 b Fy(=)f Fs(a)22 b Fy(+)h(0.)229 2711 y({)49 b(T)-8 b(ransitivit)m (\023)-46 b(e)41 b(:)f(p)s(our)g(tous)g(en)m(tier)g(naturel)h Fs(a;)17 b(b;)g(c)p Fy(,)40 b(si)g Fs(a)g Fi(6)g Fs(b)g Fy(et)g Fs(b)h Fi(6)f Fs(c)p Fy(,)g(alors)327 2831 y Fs(a)g Fi(6)h Fs(c)p Fy(.)f(En)h(e\013et)f(si,)h Fs(a)f Fi(6)h Fs(b)p Fy(,)f(on)g(p)s(eut)e(\023)-46 b(ecrire)40 b Fs(b)h Fy(=)f Fs(a)28 b Fy(+)f Fs(n)2563 2846 y Fm(1)2642 2831 y Fy(p)s(our)40 b(un)g(certain)327 2952 y(en)m(tier)c Fs(n)661 2967 y Fm(1)701 2952 y Fy(,)e(et)i(si)f Fs(b)d Fi(6)g Fs(c)p Fy(,)j(on)g(p)s(eut)d(\023)-46 b(ecrire)36 b Fs(c)c Fy(=)f Fs(b)24 b Fy(+)g Fs(n)2302 2967 y Fm(2)2376 2952 y Fy(p)s(our)35 b(un)g(certain)h(en)m(tier)327 3072 y Fs(n)385 3087 y Fm(2)425 3072 y Fy(,)e(d'o)s(\022)-51 b(u)33 b(\014nalemen)m(t)j Fs(c)30 b Fy(=)g Fs(b)23 b Fy(+)h Fs(n)1580 3087 y Fm(2)1649 3072 y Fy(=)30 b(\()p Fs(a)24 b Fy(+)f Fs(n)2025 3087 y Fm(1)2064 3072 y Fy(\))h(+)f Fs(n)2283 3087 y Fm(2)2352 3072 y Fy(=)31 b Fs(a)23 b Fy(+)g(\()p Fs(n)2728 3087 y Fm(1)2791 3072 y Fy(+)g Fs(n)2948 3087 y Fm(2)2987 3072 y Fy(\))34 b(,)g(donc)327 3193 y Fs(a)28 b Fi(6)g Fs(c)p Fy(.)229 3313 y({)49 b(An)m(tisym)m (\023)-46 b(etrie)41 b(Il)e(s'agit)g(de)h(mon)m(trer)f(que)h(si)f Fs(a)g Fi(6)g Fs(b)g Fy(et)g Fs(b)g Fi(6)g Fs(a)p Fy(,)g(alors)g Fs(a)f Fy(=)g Fs(b)p Fy(.)327 3433 y(Dans)25 b(ce)h(cas,)g(on)g(p)s (eut)d(\023)-46 b(ecrire)26 b Fs(b)i Fy(=)f Fs(a)8 b Fy(+)g Fs(n)1841 3448 y Fm(1)1880 3433 y Fy(,)26 b Fs(a)i Fy(=)f Fs(b)8 b Fy(+)g Fs(n)2306 3448 y Fm(2)2345 3433 y Fy(,)26 b(d'o)s(\022)-51 b(u)24 b Fs(a)k Fy(=)g Fs(a)8 b Fy(+)g(\()p Fs(n)3029 3448 y Fm(1)3075 3433 y Fy(+)g Fs(n)3217 3448 y Fm(2)3256 3433 y Fy(\).)327 3554 y(On)32 b(v)-5 b(a)32 b(d'ab)s(ord)g(mon)m(trer)h(que)g Fs(n)1586 3569 y Fm(1)1647 3554 y Fy(+)21 b Fs(n)1802 3569 y Fm(2)1869 3554 y Fy(=)28 b(0.)k(P)m(our)g(cel\022)-49 b(a,)33 b(on)f(v)-5 b(a)32 b(mon)m(trer)h(que)327 3674 y(p)s(our)e(tout)h Fs(b;)17 b(a;)g(b)28 b Fp(2)g Fu(N)k Fy(\()p Fs(n)20 b Fy(+)h Fs(a)28 b Fy(=)f Fs(n)21 b Fy(+)f Fs(b)p Fy(\))28 b(=)-17 b Fp(\))28 b Fy(\()p Fs(a)g Fy(=)f Fs(b)p Fy(\).)32 b(P)m(our)h(cel\022)-49 b(a,)32 b(il)g(su\016t)g(de)327 3794 y(mon)m(trer)h(que)h(p)s(our)e(tout)g Fs(n)p Fy(,)h(l'application) 1401 4028 y Fs(s)1447 4043 y Fn(n)1521 4028 y Fy(:)28 b Fu(N)83 b Fp(!)27 b Fu(N)1593 4174 y Fs(x)133 b Fp(7!)f Fs(x)23 b Fy(+)f Fs(n)327 4407 y Fy(est)53 b(injectiv)m(e.)h(Mais)e (par)g(d)m(\023)-46 b(e\014nition)53 b(de)f(l'addition,)h(on)e(a)h(la)g (form)m(ule)g(de)327 4528 y(r)m(\023)-46 b(ecurrence)30 b Fs(s)840 4543 y Fn(n)p Fm(+1)1005 4528 y Fy(=)e Fs(s)14 b Fp(\016)g Fs(s)1279 4543 y Fn(n)1325 4528 y Fy(.)28 b(Comme)i Fs(s)e Fy(est)h(injectiv)m(e)i(et)e Fs(s)2496 4543 y Fm(0)2563 4528 y Fy(aussi)h(\(c'est)f(l'iden-)327 4648 y(tit)m(\023)-46 b(e\),)38 b(il)h(est)g(facile)g(de)f(mon)m(trer)h (par)f(r)m(\023)-46 b(ecurrence)41 b(que)e Fs(s)2487 4663 y Fn(n)2572 4648 y Fy(est)g(injectiv)m(e)h(p)s(our)327 4769 y(tout)32 b Fs(n)p Fy(.)327 4889 y(Ainsi,)j(de)f Fs(a)23 b Fy(+)g(0)30 b(=)f Fs(a)h Fy(=)g Fs(a)23 b Fy(+)g(\()p Fs(n)1552 4904 y Fm(1)1615 4889 y Fy(+)g Fs(n)1772 4904 y Fm(2)1811 4889 y Fy(\),)34 b(on)g(d)m(\023)-46 b(eduit)35 b(que)f Fs(n)2592 4904 y Fm(1)2655 4889 y Fy(+)23 b Fs(n)2812 4904 y Fm(2)2881 4889 y Fy(=)30 b(0)k Fs(n)3128 4904 y Fm(1)3201 4889 y Fy(est)327 5009 y(n)m(\023)-46 b(ecessairemen)m(t)29 b(n)m(ul,)e(car)g(silon)f(il)h(existe)h Fs(k)1949 5024 y Fm(1)2014 5009 y Fy(en)m(tier)f(a)m(v)m(ec)h Fs(n)2547 5024 y Fm(1)2614 5009 y Fy(=)g Fs(s)p Fy(\()p Fs(k)2853 5024 y Fm(1)2892 5009 y Fy(\))f(=)h Fs(k)3112 5024 y Fm(1)3160 5009 y Fy(+)9 b(1.)327 5130 y(D)m(\022)-46 b(es)30 b(lors,)g(on)g(p)s(ourrait)d(\023)-46 b(ecrire)30 b(0)d(=)h Fs(n)1735 5145 y Fm(1)1791 5130 y Fy(+)17 b Fs(n)1942 5145 y Fm(2)2009 5130 y Fy(=)27 b Fs(k)2163 5145 y Fm(1)2219 5130 y Fy(+)17 b(1)g(+)g Fs(n)2529 5145 y Fm(2)2595 5130 y Fy(=)27 b(\()p Fs(k)2787 5145 y Fm(1)2843 5130 y Fy(+)17 b Fs(n)2994 5145 y Fm(2)3033 5130 y Fy(\))g(+)g(1)g(+) 327 5250 y Fs(s)p Fy(\()p Fs(k)462 5265 y Fm(1)525 5250 y Fy(+)25 b Fs(n)684 5265 y Fm(2)723 5250 y Fy(\),)36 b(ce)h(qui)f(con)m(tredirait)h(un)f(axiome)h(de)g(base)f(de)h Fu(N)p Fy(.)f(On)f(en)i(d)m(\023)-46 b(eduit)327 5370 y(que)33 b Fs(b)c Fy(=)e Fs(a)22 b Fy(+)h Fs(n)911 5385 y Fm(1)978 5370 y Fy(=)k Fs(a)c Fy(+)f(0)27 b(=)h Fs(a)p Fy(,)33 b(ce)g(qu'il)g(fallait)f(d)m(\023)-46 b(emon)m(trer.)p eop end %%Page: 6 10 TeXDict begin 6 9 bop 432 291 a @beginspecial @setspecial @endspecial Fy(6)1399 b Ft(CHAPITRE)34 b(1.)65 b(CONSTR)m(UCTION)35 b(DE)e Fu(N)578 555 y Fy(Bien)e(\023)-46 b(evidemmen)m(t,)35 b(on)d(d)m(\023)-46 b(e\014nit)34 b(la)e(relation)h Fi(>)g Fy(binaire)g(sur)g Fs(N)43 b Fy(par)1328 758 y Fp(8)p Fs(a;)17 b(b)28 b Fp(2)g Fu(N)1713 717 y Fm(2)1850 758 y Fy(\()p Fs(a)g Fi(>)g Fs(b)p Fy(\))56 b Fp(\()-17 b(\))55 b Fy(\()p Fs(b)28 b Fi(6)g Fs(a)p Fy(\))p Fs(:)432 961 y Fy(Il)33 b(est)g(facile)g(de)g(d)m(\023)-46 b(eduire)34 b(de)f(ce)g(qui)g(pr)m(\023)-46 b(ec)m(\022)g(ede)35 b(que)e Fi(>)g Fy(est)g(une)h(relation)e(d'ordre.)432 980 y @beginspecial @setspecial @endspecial 170 x Fx(Th)n(\023)-54 b(eor)n(\022)g(eme)42 b(1.)i Fy(\()p Fu(N)p Fs(;)17 b Fi(6)p Fy(\))38 b Fz(est)h(un)f(ensemble)f(totalement)h(or)-5 b(donn)n(\023)-47 b(e,)36 b(c'est)i(\022)-50 b(a)39 b(dir)-5 b(e)38 b(que)432 1270 y(p)-5 b(our)35 b(tout)g Fy(\()p Fs(a;)17 b(b)p Fy(\))28 b Fp(2)g Fu(N)1262 1234 y Fm(2)1337 1270 y Fz(on)34 b(a)h Fs(a)28 b Fp(\024)g Fs(b)35 b Fz(ou)g Fs(b)28 b Fp(\024)h Fs(a)p Fz(.)432 1459 y(D)n(\023)-47 b(emonstr)-5 b(ation.)46 b Fy(On)33 b(v)-5 b(a)32 b(prouv)m(er)i(cela)f (par)f(r)m(\023)-46 b(ecurrence)35 b(:)d(soit)1268 1662 y Fs(C)i Fy(=)28 b Fp(f)p Fs(a)f Fp(2)h Fu(N)p Fy(;)17 b Fp(8)p Fs(b)29 b Fp(2)f Fs(N)10 b Fy(;)17 b Fs(a)28 b Fp(\024)g Fs(b)33 b Fy(ou)f Fs(b)d Fp(\024)f Fs(a)p Fp(g)432 1865 y Fy(Il)37 b(est)i(clair)e(que)i(0)c Fp(2)i Fs(C)7 b Fy(,)37 b(car)h Fs(b)e Fy(=)g(0)25 b(+)g Fs(b)p Fy(,)38 b(soit)g(0)e Fp(\024)g Fs(b)i Fy(p)s(our)f(tout)h(en)m(tier)g (naturel)g Fs(b)p Fy(.)432 1985 y(Supp)s(osons)j Fs(a)h Fp(2)f Fs(C)48 b Fy(et)40 b(mon)m(trons)i(que)f Fs(a)28 b Fy(+)f(1)41 b Fp(2)h Fs(C)7 b Fy(.)40 b(Soit)h Fs(n)g Fp(2)g Fu(N)g Fy(Si)g Fs(b)g Fy(=)g(0,)f(on)h(a)429 2105 y(\023)-46 b(evidemmen)m(t)30 b Fs(b)e Fp(\024)g Fs(a)11 b Fy(+)g(1.)28 b(Sinon,)g(il)f(existe)i Fs(b)2085 2069 y Fo(0)2136 2105 y Fy(en)m(tier)g(tel)e(que)h Fs(b)h Fy(=)e Fs(b)2930 2069 y Fo(0)2965 2105 y Fy(+)11 b(1.)28 b(L'h)m(yp)s(oth)m(\022)-46 b(ese)432 2226 y(de)33 b(r)m(\023)-46 b(ecurrence)34 b(nous)f(dit)g(que)h Fs(b)1636 2190 y Fo(0)1687 2226 y Fp(\024)28 b Fs(a)33 b Fy(ou)g Fs(b)2053 2190 y Fo(0)2104 2226 y Fp(\025)28 b Fs(a)p Fy(.)33 b(Il)g(y)g(a)f (deux)i(cas)f(p)s(ossibles)578 2346 y({)49 b(si)33 b Fs(b)815 2310 y Fo(0)866 2346 y Fp(\024)28 b Fs(a)p Fy(,)33 b(alors)g Fs(b)28 b Fy(=)f Fs(b)1529 2310 y Fo(0)1575 2346 y Fy(+)22 b(1)28 b Fp(\024)g Fs(a)22 b Fy(+)g(1.)578 2467 y({)49 b(si)35 b Fs(a)d Fp(\024)g Fs(b)1009 2430 y Fo(0)1067 2467 y Fy(et)j Fs(a)d Fp(6)p Fy(=)f Fs(b)1414 2430 y Fo(0)1438 2467 y Fy(,)k(alors)g(on)g(p)s(eut)d(\023)-46 b(ecrire)36 b Fs(b)2410 2430 y Fo(0)2465 2467 y Fy(=)31 b Fs(a)24 b Fy(+)g Fs(c)34 b Fy(a)m(v)m(ec)j Fs(c)31 b Fp(6)p Fy(=)h(0,)i(donc)i(on)676 2587 y(p)s(eut)26 b(\023)-46 b(ecrire)29 b Fs(c)f Fy(=)g Fs(c)1374 2551 y Fo(0)1411 2587 y Fy(+)14 b(1,)29 b(d'o)s(\022)-51 b(u)28 b Fs(b)1860 2551 y Fo(0)1911 2587 y Fy(=)g Fs(a)14 b Fy(+)g Fs(c)2212 2551 y Fo(0)2250 2587 y Fy(+)g(1)28 b(=)g(\()p Fs(a)14 b Fy(+)g(1\))g(+)g Fs(c)2947 2551 y Fo(0)2971 2587 y Fy(,)29 b(d'o)s(\022)-51 b(u)28 b Fs(a)14 b Fy(+)g(1)28 b Fp(\024)g Fs(c)3619 2551 y Fo(0)3642 2587 y Fy(.)p 3599 2707 4 66 v 3603 2645 59 4 v 3603 2707 V 3661 2707 4 66 v 578 2898 a(On)33 b(note)1499 3018 y(\()p Fs(a)28 b(<)g(b)p Fy(\))g(=)f(\()p Fs(a)h Fi(6)g Fs(b)p Fy(\))33 b(et)g Fs(a)28 b Fp(6)p Fy(=)f Fs(b)432 3185 y Fy(ainsi)33 b(que)1499 3305 y(\()p Fs(a)28 b(>)g(b)p Fy(\))g(=)f(\()p Fs(a)h Fi(>)g Fs(b)p Fy(\))33 b(et)g Fs(a)28 b Fp(6)p Fy(=)f Fs(b)578 3472 y Fy(En)f(utilisan)m(t)h (le)f(fait)f(que)i(l'ordre)f(est)g(total,)g(il)f(est)i(facile)f(de)g(v) m(oir)g(que)h(le)f(con)m(traire)432 3593 y(de)33 b Fs(a)28 b(<)f(b)33 b Fy(\()p Fz(r)-5 b(esp.)35 b Fs(a)27 b(>)h(b)p Fy(\))33 b(est)g Fs(a)28 b Fp(\025)g Fs(b)33 b Fy(\()p Fz(r)-5 b(esp.)34 b Fs(a)28 b Fp(\024)h Fs(b)p Fy(\).)578 3713 y(Le)e(lemme)g(qui)g(suit)g(est)h(utilis)m(\023)-46 b(e)27 b(tr)m(\022)-46 b(es)27 b(fr)m(\023)-46 b(equemmen)m(t)29 b(dans)e(les)g(raisonnemen)m(ts)i(qui)432 3834 y(metten)m(t)34 b(en)f(o)s(euvre)g(des)g(en)m(tiers)i(:)-1318 b @beginspecial @setspecial @endspecial 188 x Fx(Lemme)38 b(1.)k Fz(Pour)35 b(tous)h(entiers)e Fs(n)h Fz(et)g Fs(p)p Fz(,)g(on)f(a)578 4143 y({)1627 4263 y Fy(\()p Fs(n)28 b(<)f(p)p Fy(\))56 b Fp(\()-17 b(\))55 b Fy(\()p Fs(n)22 b Fy(+)g(1)27 b Fp(\024)h Fs(p)p Fy(\))578 4430 y Fz({)1627 4550 y Fy(\()p Fs(n)g Fp(\025)g Fs(p)p Fy(\))55 b Fp(\()-17 b(\))55 b Fy(\()p Fs(n)22 b Fy(+)g(1)28 b Fs(>)f(p)p Fy(\))676 4739 y Fz(D)n(\023)-47 b(emonstr)-5 b(ation.)46 b Fy(On)33 b(v)-5 b(a)33 b(seulemen)m(t)j(prouv)m(er)f(la)e(premi)m(\022)-46 b(ere)32 b(\023)-46 b(equiv)-5 b(alence,)36 b(car)676 4860 y(la)c(deuxi)m(\022)-46 b(eme)35 b(n'est)e(que)h(la)e(con)m(trap)s (os)m(\023)-46 b(ee)34 b(de)f(la)f(premi)m(\022)-46 b(ere.)676 5009 y(Sens)40 b(direct.)g(Supp)s(osons)g Fs(n)g(<)f(p)g Fy(:)g(on)g(a)g(donc)h Fs(n)f Fp(\024)g Fs(p)g Fy(:)h(il)f(existe)i(un) e(en)m(tier)676 5130 y(naturel)34 b Fs(k)i Fy(a)m(v)m(ec)f Fs(p)29 b Fy(=)g Fs(n)23 b Fy(+)g Fs(k)s Fy(.)33 b Fs(k)g Fp(6)p Fy(=)c(0,)k(sinon)h(on)g(aurait)f Fs(n)c Fy(=)g Fs(p)34 b Fy(:)f(il)h(existe)h(donc)676 5250 y(un)h(en)m(tier)g Fs(m)g Fy(tel)g(que)h Fs(k)f Fy(=)c Fs(m)25 b Fy(+)f(1)35 b(:)h(on)g(a)f(donc)h Fs(p)d Fy(=)f Fs(n)25 b Fy(+)f Fs(k)36 b Fy(=)d(\()p Fs(n)24 b Fy(+)g(1\))g(+)g Fs(m)p Fy(,)676 5370 y(d'o)s(\022)-51 b(u)31 b Fs(n)23 b Fy(+)f(1)27 b Fp(\024)h Fs(p)p Fy(.)p eop end %%Page: 7 11 TeXDict begin 7 10 bop 83 291 a @beginspecial @setspecial @endspecial Ft(1.4.)65 b Fu(N)p Ft(,)32 b(CE)i(MONO)912 265 y(\177)915 291 y(IDE)2181 b Fy(7)229 555 y({)-49 b({)49 b(R)m(\023)-46 b(ecipro)s(que.)42 b(Supp)s(osons)f Fs(n)28 b Fy(+)f(1)41 b Fp(\024)h Fs(p)e Fy(:)g(il)h(existe)h(un)f(en)m (tier)g Fs(k)j Fy(tel)d(que)g Fs(p)g Fy(=)327 676 y(\()p Fs(n)17 b Fy(+)g(1\))g(+)g Fs(k)30 b Fy(=)e Fs(n)17 b Fy(+)g(\()p Fs(k)j Fy(+)d(1\).)30 b(On)g(a)f(donc)i Fs(n)d Fp(\024)g Fs(p)p Fy(.)i(Mon)m(trons)h(que)g Fs(n)d Fp(6)p Fy(=)g Fs(p)p Fy(.)i(Si)g(on)327 796 y(a)m(v)-5 b(ait)33 b Fs(n)28 b Fy(=)g Fs(p)p Fy(,)33 b(on)g(aurait)f Fs(n)23 b Fy(+)f(0)28 b(=)g Fs(n)h Fy(=)f Fs(n)22 b Fy(+)h(\()p Fs(k)i Fy(+)d(1\),)33 b(d'o)s(\022)-51 b(u)32 b Fs(k)25 b Fy(+)e(1)k(=)i Fs(s)p Fy(\()p Fs(k)s Fy(\)0,)j(ce)327 916 y(qui)h(est)g(imp)s(ossible.)p 3250 1052 4 66 v 3254 990 59 4 v 3254 1052 V 3312 1052 4 66 v 83 1252 a @beginspecial @setspecial @endspecial 82 x Fx(Corollaire)57 b(1.)50 b Fz(Pour)g(tout)g Fs(n)56 b Fp(2)f Fu(N)p Fz(,)50 b(l'unique)d(\023) -47 b(el)n(\023)g(exent)48 b Fs(x)i Fz(qui)g(satisfasse)e(\014mul-)83 1454 y(tan)n(\023)-47 b(ement)33 b(aux)i(in)n(\023)-47 b(equations)1298 1721 y Fs(n)28 b Fp(\024)g Fs(x)36 b Fz(et)f Fs(x)28 b(<)f(n)c Fy(+)f(1)83 1987 y Fz(est)35 b Fs(x)28 b Fy(=)g Fs(n)p Fz(.)83 2269 y(D)n(\023)-47 b(emonstr)-5 b(ation.)46 b Fy(Mon)m(trons)41 b(que)f(que)g Fs(x)g Fy(=)f Fs(n)g Fy(est)i(la)e(seule)h(solution)g(p)s(ossible)h(:)e (si)83 2389 y Fs(x)28 b(<)g(n)13 b Fy(+)g(1,)28 b(on)g(a)f Fs(x)i Fp(\024)f Fs(n)p Fy(,)g(d'apr)m(\022)-46 b(es)29 b(la)f(deuxi)m(\022)-46 b(eme)28 b(\023)-46 b(equiv)-5 b(alence)30 b(du)e(lemme)i(pr)m(\023)-46 b(ec)m(\023)g(eden)m(t)30 b(:)83 2509 y(ainsi)j Fs(n)28 b Fp(\024)g Fs(x)g(<)g(n)22 b Fy(+)g(1)33 b(implique)h Fs(n)28 b Fp(\024)g Fs(x)g Fp(\024)g Fs(n)33 b Fy(d'o)s(\022)-51 b(u)32 b Fs(x)c Fy(=)f Fs(n)p Fy(.)229 2645 y(V)m(\023)-46 b(eri\014ons)42 b(main)m(tenan)m(t)g(que)f Fs(x)h Fy(=)e Fs(n)h Fy(est)g(bien)g(une)h (solution.)f(On)f(doit)h(v)m(\023)-46 b(eri\014er)83 2766 y Fs(n)29 b Fp(\024)f Fs(n)34 b Fy(et)f Fs(n)28 b(<)h(n)22 b Fy(+)h(1)32 b(:)h(la)g(premi)m(\022)-46 b(ere)34 b(in)m(\023)-46 b(egalit)m(\023)g(e)34 b(est)d(\023)-46 b(eviden)m(te,)35 b(quan)m(t)f(\022)-49 b(a)32 b(la)h(premi)m(\022)-46 b(ere,)83 2886 y(c'est)39 b(une)g(cons)m(\023)-46 b(equence)42 b(du)c(sens)i(\(2\))d(=)-17 b Fp(\))37 b Fy(\(1\))h(dans)g(la)g(premi)m (\022)-46 b(ere)37 b(\023)-46 b(equiv)-5 b(alence)40 b(du)83 3007 y(lemme)34 b(pr)m(\023)-46 b(ec)m(\023)g(eden)m(t.)p 3250 3007 V 3254 2944 59 4 v 3254 3007 V 3312 3007 4 66 v 83 3225 a @beginspecial @setspecial @endspecial 63 x Fx(Th)n(\023)-54 b(eor)n(\022)g(eme)52 b(2.)c Fz(L'ensemble)d(or) -5 b(donn)n(\023)-47 b(e)44 b Fy(\()p Fu(N)p Fs(;)17 b Fi(6)p Fy(\))46 b Fz(est)g(bien)f(or)-5 b(donn)n(\023)-47 b(e)44 b(:)i(toute)h(p)-5 b(artie)83 3408 y(non)34 b(vide)h(de)f Fu(N)h Fz(admet)f(un)h(plus)g(p)-5 b(etit)33 b(\023)-47 b(el)n(\023)g(ement.)83 3690 y(D)n(\023)g(emonstr)-5 b(ation.)46 b Fy(Soit)35 b Fs(A)g Fy(une)g(partie)g(non)g(vide)g(de)h Fu(N)p Fy(.)e(Notons)h Fs(M)45 b Fy(l'ensem)m(ble)38 b(des)83 3810 y(minoran)m(ts)f(de)g Fs(A)p Fy(.)g Fs(M)47 b Fy(est)37 b(non-vide)g(car)f(0)e Fp(2)g Fs(M)10 b Fy(.)37 b(Il)g(est)g(clair)g(que)g Fs(M)47 b Fy(n'est)37 b(pas)g Fu(N)83 3930 y Fy(tout)31 b(en)m(tier,)h(car)f(si)h Fs(a)c Fp(2)g Fs(A)p Fy(,)j Fs(a)20 b Fy(+)f(1)27 b Fs(>)h(a)p Fy(,)j(donc)p Fs(a)20 b Fy(+)f(1)39 b Fs(=)-61 b Fp(2)28 b Fs(M)10 b Fy(.)32 b(Il)g(n'est)g(pas)f(p)s(ossible)i(que)83 4051 y(l'implication)39 b(\()p Fs(a)e Fp(2)g Fs(M)10 b Fy(\))38 b(=)-17 b Fp(\))37 b Fy(\()p Fs(a)26 b Fy(+)f(1)37 b Fp(2)g Fs(M)10 b Fy(\),)39 b(car)f(comme)h(0)e Fp(2)g Fs(M)10 b Fy(,)39 b(le)f(princip)s(e)h(de)83 4171 y(r)m(\023)-46 b(ecurrence)31 b(impliquerait)f(que)g Fs(M)39 b Fy(=)27 b Fu(N)p Fy(,)i(ce)h(qui,)g(on)f(l'a)g(vu,)g(est)h(faux.)f(Il)h(existe) g(donc)83 4291 y Fs(a)134 4306 y Fm(0)211 4291 y Fp(2)38 b Fs(M)49 b Fy(tel)39 b(que)g Fs(a)843 4306 y Fm(0)909 4291 y Fy(+)26 b(1)49 b Fs(=)-61 b Fp(2)38 b Fs(M)10 b Fy(.)39 b(Si)g Fs(a)1542 4306 y Fm(0)1608 4291 y Fy(+)26 b(1)49 b Fs(=)-61 b Fp(2)38 b Fs(M)10 b Fy(,)39 b(c'est)h(qu'il)f (existe)h Fs(x)e Fp(2)g Fs(A)p Fy(,)g(a)m(v)m(ec)83 4412 y Fs(x)29 b(<)g(a)323 4427 y Fm(0)385 4412 y Fy(+)23 b(1.)33 b(Mais)h(comme)g Fs(a)1212 4427 y Fm(0)1285 4412 y Fy(minore)f Fs(a)p Fy(,)h(on)f(a)g Fs(a)1991 4427 y Fm(0)2059 4412 y Fp(\024)d Fs(x)p Fy(,)j(d'o)s(\022)-51 b(u)33 b Fs(a)2550 4427 y Fm(0)2618 4412 y Fp(\024)d Fs(x)f(<)f(a)2964 4427 y Fm(0)3027 4412 y Fy(+)22 b(1,)33 b(ce)83 4532 y(qui)38 b(implique)i Fs(x)c Fy(=)g Fs(a)911 4547 y Fm(0)951 4532 y Fy(,)i(donc)g Fs(a)1305 4547 y Fm(0)1380 4532 y Fp(2)f Fs(A)p Fy(.)h(Ainsi)g Fs(a)1929 4547 y Fm(0)2007 4532 y Fy(est)g(dans)g Fs(A)g Fy(est)g(minore)g(tous)g (les)80 4653 y(\023)-46 b(el)m(\023)g(emen)m(ts)35 b(de)e Fs(A)g Fy(:)f(c'est)i(son)f(plus)g(p)s(etit)d(\023)-46 b(el)m(\023)g(emen)m(t.)p 3250 4789 V 3254 4726 59 4 v 3254 4789 V 3312 4789 4 66 v 83 4988 a @beginspecial @setspecial @endspecial 82 x Fx(Th)n(\023)-54 b(eor)n(\022)g(eme)38 b(3.)k Fz(T)-7 b(oute)34 b(p)-5 b(artie)35 b(major)n(\023)-47 b(ee)33 b(de)h Fu(N)h Fz(admet)g(un)g(plus)f(gr)-5 b(and)32 b(\023)-47 b(el)n(\023)g(ement.)83 5351 y(D)n(\023)g(emonstr)-5 b(ation.)46 b Fy(La)33 b(preuv)m(e)h(est)f(laiss)m(\023)-46 b(ee)34 b(en)g(exercice.)p 3250 5351 V 3254 5289 59 4 v 3254 5351 V 3312 5351 4 66 v eop end %%Page: 8 12 TeXDict begin 8 11 bop 432 291 a @beginspecial @setspecial @endspecial Fy(8)1399 b Ft(CHAPITRE)34 b(1.)65 b(CONSTR)m(UCTION)35 b(DE)e Fu(N)432 456 y @beginspecial @setspecial @endspecial 99 x Fq(1.5)160 b(Multiplication)52 b(des)h(en)l(tiers)432 682 y @beginspecial @setspecial @endspecial 121 x Fw(1.5.1)136 b(D)m(\023)-64 b(e\014nition)45 b(et)h(propri)m(\023)-64 b(et)m(\023)g(es)432 891 y @beginspecial @setspecial @endspecial 97 x Fx(Th)n(\023)-54 b(eor)n(\022)g(eme)27 b(4)g(\(Admis\).)35 b Fz(Il)26 b(existe)g(une)g(unique)h(loi)f(de)g(c) -5 b(omp)g(osition)25 b(interne)h(not)n(\023)-47 b(ee)432 1108 y Fp(\002)35 b Fz(sur)g Fu(N)g Fz(v)n(\023)-47 b(eri\014ant)33 b(:)2076 1323 y Fp(8)p Fs(n)28 b Fp(2)g Fu(N)100 b Fs(n)22 b Fp(\002)h Fy(0)k(=)h(0)1044 1468 y Fp(8)p Fy(\()p Fs(n;)17 b(p)p Fy(\))27 b Fp(2)h Fu(N)23 b Fp(\002)f Fu(N)100 b Fs(p)22 b Fp(\002)g Fy(\()p Fs(n)h Fy(+)f(1\))27 b(=)h(\()p Fs(p)22 b Fp(\002)g Fs(n)p Fy(\))h(+)f Fs(p)432 1682 y Fz(On)34 b(l'app)-5 b(el)5 b(le)34 b(multiplic)-5 b(ation)432 1780 y @beginspecial @setspecial @endspecial 101 x Fx(Prop)s(osition)37 b(1)h(\(Propri)n(\023)-54 b(et)n(\023)g(es)36 b(de)i(la)g(m)m (ultiplication\).)189 b Fz({)48 b(Asso)-5 b(ciatit)n(\023)-47 b(e)24 b(:)h Fp(8)p Fy(\()p Fs(p;)17 b(q)t(;)g(r)s Fy(\))27 b Fp(2)677 2001 y Fu(N)749 1965 y Fm(3)888 2001 y Fy(\()p Fs(p)22 b Fp(\002)g Fs(q)t Fy(\))g Fp(\002)h Fs(r)30 b Fy(=)e Fs(p)22 b Fp(\002)g Fy(\()p Fs(q)k Fp(\002)d Fs(r)s Fy(\))p Fz(.)578 2121 y({)49 b(Distributivit)n(\023)-47 b(e)37 b(p)-5 b(ar)38 b(r)-5 b(app)g(ort)38 b(\022)-50 b(a)38 b(l'addition)f(:)h Fp(8)p Fy(\()p Fs(p;)17 b(q)t(;)g(r)s Fy(\))33 b Fp(2)h Fu(N)2951 2085 y Fm(3)3090 2121 y Fy(\()p Fs(p)24 b Fy(+)h Fs(q)t Fy(\))f Fp(\002)h Fs(r)37 b Fy(=)677 2242 y Fs(p)22 b Fp(\002)g Fs(r)j Fy(+)d Fs(q)k Fp(\002)d Fs(r)s Fz(.)34 b(et)h Fp(8)p Fy(\()p Fs(p;)17 b(q)t(;)g(r)s Fy(\))27 b Fp(2)h Fu(N)1961 2206 y Fm(3)2100 2242 y Fs(r)d Fp(\002)e Fy(\()p Fs(p)f Fy(+)g Fs(q)t Fy(\))g Fp(\002)g Fs(r)30 b Fy(=)e Fs(r)d Fp(\002)d Fs(p)h Fy(+)f Fs(r)i Fp(\002)f Fs(q)t Fz(.)578 2362 y({)49 b(Commutativit)n(\023)-47 b(e)33 b Fp(8)p Fy(\()p Fs(p;)17 b(q)t Fy(\))28 b Fp(2)g Fu(N)1804 2326 y Fm(2)1943 2362 y Fs(p)22 b Fp(\002)g Fs(q)32 b Fy(=)27 b Fs(q)f Fp(\002)d Fs(p)p Fz(.)578 2482 y({)49 b(R)n(\023)-47 b(egularit)n(\023)g(e)32 b(de)j(la)g (multiplic)-5 b(ation)34 b(p)-5 b(ar)34 b(un)h(entier)g(non)f(nul)h(:) 1315 2697 y Fp(8)p Fy(\()p Fs(p;)17 b(q)t(;)g(r)s Fy(\))27 b Fp(2)h Fu(N)1870 2656 y Fm(2)1932 2697 y Fp(\002)22 b Fu(N)2103 2712 y Fo(\003)2242 2697 y Fs(pr)31 b Fy(=)c Fs(q)t(r)j Fy(=)-17 b Fp(\))28 b Fs(p)f Fy(=)h Fs(q)t(:)578 2911 y Fy(Remarque)34 b(:)1067 3152 y(\()p Fs(p)22 b Fp(\002)h Fy(1\))k(=)h Fs(p)22 b Fp(\002)g Fy(\(0)g(+)g(1\))28 b(=)f Fs(p)22 b Fp(\002)h Fy(0)f(+)g Fs(p)27 b Fy(=)h(0)22 b(+)g Fs(p)28 b Fy(=)f Fs(p:)432 3315 y @beginspecial @setspecial @endspecial 125 x Fw(1.5.2)136 b(Division)46 b(euclidienne)432 3504 y @beginspecial @setspecial @endspecial 120 x Fx(Lemme)41 b(2)e(\(Propri)n(\023)-54 b(et)n(\023)g(e)39 b(d'Arc)m(him)n(\022)-54 b(ede\).)43 b Fz(Si)37 b Fy(\()p Fs(a;)17 b(b)p Fy(\))31 b Fp(2)h Fu(N)23 b Fp(\002)h Fu(N)2940 3639 y Fo(\003)2979 3624 y Fz(,)37 b(il)f(existe)h Fs(n)31 b Fp(2)h Fu(N)432 3745 y Fz(tel)j(que)g Fs(nb)28 b(>)f(a)p Fz(.)432 3943 y(D)n(\023)-47 b(emonstr)-5 b(ation.)46 b Fy(Comme)36 b Fs(b)c Fp(\025)g Fy(1,)j(et)g(on)g(p)s(eut)g(prendre)h (par)f(exemple)i Fs(n)32 b Fy(=)f Fs(a)24 b Fy(+)g(1)34 b(:)432 4064 y(\()p Fs(a)22 b Fy(+)g(1\))p Fs(b)28 b Fp(\025)g Fs(a)22 b Fy(+)g(1)28 b Fs(>)f(a)p Fy(.)p 3599 4064 4 66 v 3603 4001 59 4 v 3603 4064 V 3661 4064 4 66 v 432 4162 a @beginspecial @setspecial @endspecial 100 x Fx(Th)n(\023)-54 b(eor)n(\022)g(eme)40 b(5.)i Fz(Si)37 b Fy(\()p Fs(a;)17 b(b)p Fy(\))30 b Fp(2)h Fu(N)23 b Fp(\002)h Fu(N)1818 4277 y Fo(\003)1858 4262 y Fz(.)36 b(Il)g(existe)g(un)g(unique)h(c)-5 b(ouple)36 b Fy(\()p Fs(q)t(;)17 b(r)s Fy(\))35 b Fz(d'entiers)432 4382 y(natur)-5 b(els)35 b(v)n(\023)-47 b(eri\014ant)33 b Fs(a)28 b Fy(=)f Fs(nq)g Fy(+)22 b Fs(r)s Fz(.)432 4581 y(D)n(\023)-47 b(emonstr)-5 b(ation.)192 b Fy({)49 b(Mon)m(trons)43 b(d'ab)s(ord)f(l'existence)j(d'un)e(tel)f(couple.)i(On)e(v)-5 b(a)676 4701 y(mon)m(trer)33 b(par)f(r)m(\023)-46 b(ecurrence)35 b(sur)e Fs(p)f Fy(la)h(propri)m(\023)-46 b(et)m(\023)g(e)33 b(:)865 4915 y(\()p Fs(H)984 4930 y Fn(p)1024 4915 y Fy(\))28 b(:)f(\()p Fs(n)h(<)g(bp)p Fy(\))g(=)-17 b Fp(\))27 b(9)p Fy(\()p Fs(q)t(;)17 b(r)s Fy(\))27 b Fp(2)h Fu(N)23 b Fp(\002)f Fu(N)98 b Fs(a)28 b Fy(=)f Fs(bq)g Fy(+)22 b Fs(r)35 b Fy(et)e(0)27 b Fp(\024)h Fs(r)j(<)c(b:)676 5130 y Fy(P)m(our)41 b Fs(p)g Fy(=)g(0)f(c'est)i(vrai)f(car)f(le)h (faux)g(implique)i(le)e(vrai.)f(P)m(our)i Fs(p)f Fy(=)g(1,)f(c'est)676 5250 y(vrai)45 b(car)f(si)i Fs(n)i(<)g(p)p Fy(,)d(on)d(\023)-46 b(ecrit)45 b Fs(n)k Fy(=)f(0)p Fs(:b)31 b Fy(+)f Fs(n)p Fy(.)45 b(Mon)m(trons)h(que)f(la)g(propri)m(\023)-46 b(et)m(\023)g(e)676 5370 y(est)42 b(h)m(\023)-46 b(er)m(\023)g (editaire)42 b(:)g(Soit)f Fs(n)i(<)g(b)p Fy(\()p Fs(p)29 b Fy(+)f(1\).)41 b(Si)h Fs(n)h(<)f(b)g Fy(la)g(preuv)m(e)h(est)f (termin)m(\023)-46 b(ee.)p eop end %%Page: 9 13 TeXDict begin 9 12 bop 83 291 a @beginspecial @setspecial @endspecial Ft(1.5.)65 b(MUL)-8 b(TIPLICA)g(TION)35 b(DES)d(ENTIERS) 1403 b Fy(9)327 555 y(Sinon)37 b Fs(n)f Fp(\025)f Fs(b)p Fy(,)j(donc)f(il)g(exisxte)j(un)d(en)m(tier)h Fs(n)2054 570 y Fm(1)2131 555 y Fy(tel)f(que)h Fs(n)d Fy(=)g Fs(n)2724 570 y Fm(1)2789 555 y Fy(+)25 b Fs(b)p Fy(.)38 b(Comme)327 676 y Fs(n)c(<)f(b)p Fy(\()p Fs(p)25 b Fy(+)f(1\),)36 b(on)g(a)g(n')m(\023)-46 b(ecessairemen)m(t)39 b Fs(n)1908 691 y Fm(1)1981 676 y Fs(<)34 b(bp)p Fy(.)i(Donc)g(d'apr)m(\022)-46 b(es)37 b(l'h)m(yp)s(oth)m(\022)-46 b(ese)327 796 y(de)33 b(r)m(\023)-46 b(ecurrence,)35 b(on)e(p)s(eut)d(\023)-46 b(ecrire,)33 b Fs(n)1667 811 y Fm(1)1735 796 y Fy(=)28 b Fs(bq)e Fy(+)d Fs(r)35 b Fy(a)m(v)m(ec)f Fs(q)e Fp(2)c Fu(N)33 b Fy(et)g(0)28 b Fp(\024)g Fs(r)j(<)d(b)p Fy(.)33 b(On)327 916 y(en)h(d)m(\023)-46 b(eduit)34 b(l')m(\023)-46 b(ecriture)35 b Fs(n)30 b Fy(=)f(\()p Fs(b)23 b Fy(+)g(1\))p Fs(q)j Fy(+)d Fs(r)s Fy(.)33 b(Ainsi)i(\()p Fs(H)2304 931 y Fn(p)2343 916 y Fy(\))f(est)g(r)m(\023)-46 b(ealis)m(\023)g(ee)35 b(p)s(our)e(tout)327 1037 y Fs(p)p Fy(.)c(Soit)g(main)m(tenan)m(t)h Fs(n)f Fy(en)m(tier)h(:)f(d'apr)m(\022)-46 b(es)30 b(la)e(propri)m (\023)-46 b(et)m(\023)g(e)30 b(d'Arc)m(him)m(\022)-46 b(ede,)31 b(il)e(existe)327 1157 y Fs(p)39 b Fy(tel)h(que)g Fs(n)g(<)f(bp)h Fy(:)g(comme)g Fs(H)1579 1172 y Fn(p)1658 1157 y Fy(est)g(vraie,)g(il)g(existe)h(\()p Fs(q)t(;)17 b(r)s Fy(\))38 b Fp(2)i Fu(N)27 b Fp(\002)g Fu(N)39 b Fy(a)m(v)m(ec)327 1277 y Fs(a)28 b Fy(=)f Fs(bq)g Fy(+)22 b Fs(r)35 b Fy(et)e(0)27 b Fp(\024)h Fs(r)j(<)c(b)p Fy(.)229 1398 y({)49 b(Mon)m(trons)34 b(main)m(tenan)m(t)h(l'unicit)m(\023)-46 b(e)34 b(:)g(supp)s(osons)g(que)g(l'on)g(ait)f Fs(a)c Fy(=)g Fs(bq)2972 1413 y Fm(1)3034 1398 y Fy(+)23 b Fs(r)3177 1413 y Fm(1)3245 1398 y Fy(=)327 1518 y Fs(bq)411 1533 y Fm(2)480 1518 y Fy(+)28 b Fs(r)628 1533 y Fm(2)668 1518 y Fy(.)42 b(Quitte)h(\022)-49 b(a)39 b(\023)-46 b(ec)m(hanger)43 b(les)h(roles,)f(on)f(p)s(eut)h(supp)s(oser)g(que)g Fs(q)3005 1533 y Fm(1)3089 1518 y Fp(\024)i Fs(q)3254 1533 y Fm(2)3294 1518 y Fy(,)327 1639 y(ainsi)35 b(on)g(p)s(eut)d(\023) -46 b(ecrire)35 b Fs(q)1232 1654 y Fm(2)1303 1639 y Fy(=)c Fs(q)1453 1654 y Fm(1)1517 1639 y Fy(+)23 b Fs(d)p Fy(,)35 b(a)m(v)m(ec)h Fs(d)31 b Fp(2)g Fu(N)p Fy(,)k(d'o)s(\022)-51 b(u)34 b(il)h(ressort)g Fs(bq)2967 1654 y Fm(1)3031 1639 y Fy(+)23 b Fs(r)3174 1654 y Fm(1)3245 1639 y Fy(=)327 1759 y Fs(bq)411 1774 y Fm(1)479 1759 y Fy(+)k Fs(bd)h Fy(+)g Fs(r)850 1774 y Fm(2)889 1759 y Fy(,)41 b(d'o)s(\022)-51 b(u)40 b Fs(r)1226 1774 y Fm(1)1307 1759 y Fy(=)i Fs(bd)28 b Fy(+)f Fs(r)1692 1774 y Fm(2)1732 1759 y Fy(.)41 b(On)f(a)h Fs(bd)h Fp(\024)g Fs(bd)28 b Fy(+)g Fs(r)2581 1774 y Fm(2)2662 1759 y Fy(=)41 b Fs(r)2823 1774 y Fm(1)2904 1759 y Fs(<)h(b)p Fy(.)f(Cela)327 1879 y(implique)34 b(que)g Fs(d)27 b Fy(=)h(0.)k(On)h(en)g(d)m(\023)-46 b(eduit)33 b Fs(r)1837 1894 y Fm(1)1904 1879 y Fy(=)28 b Fs(r)2052 1894 y Fm(2)2124 1879 y Fy(et)33 b Fs(q)2281 1894 y Fm(1)2348 1879 y Fy(=)27 b Fs(q)2494 1894 y Fm(2)2534 1879 y Fy(.)p 3250 2000 4 66 v 3254 1937 59 4 v 3254 2000 V 3312 2000 4 66 v 83 2136 a @beginspecial @setspecial @endspecial 150 x Fw(1.5.3)136 b(Bases)45 b(de)g(n)l(um)m(\023)-64 b(eration)83 2350 y @beginspecial @setspecial @endspecial 120 x Fx(Th)n(\023)-54 b(eor)n(\022)g(eme)39 b(6.)k Fz(Soit)35 b Fs(b)i Fz(un)f(entier)f(natur)-5 b(el)36 b(non)g(nul.)f(Pour)h(tout)h (entier)f(natur)-5 b(el)36 b Fs(n)p Fz(,)83 2591 y(on)f(p)-5 b(eut)36 b(tr)-5 b(ouver)36 b(un)g(entier)g(natur)-5 b(el)36 b Fs(p)f Fz(et)h(une)g(suite)g(suite)g(d'entiers)f Fy(\()p Fs(a)2875 2606 y Fn(n)2922 2591 y Fy(\))2960 2606 y Fm(0)p Fo(\024)p Fn(k)r Fo(\024)p Fn(p)3219 2591 y Fz(tel)83 2711 y(que)1446 2888 y Fs(n)27 b Fy(=)1682 2776 y Fn(p)1639 2806 y Fk(X)1635 2989 y Fn(k)r Fm(=0)1780 2888 y Fs(a)1831 2903 y Fn(k)1874 2888 y Fs(b)1915 2847 y Fn(k)83 3093 y Fz(ave)-5 b(c)644 3214 y Fp(8)p Fs(k)31 b Fp(2)d(f)p Fy(0)p Fs(;)17 b(:)g(:)g(:)e(;)i(p)p Fp(g)99 b Fy(0)28 b Fp(\024)g Fs(a)1623 3229 y Fn(k)1694 3214 y Fs(<)f(b)35 b Fz(et)h Fy(\()p Fs(a)2075 3229 y Fn(p)2142 3214 y Fs(>)28 b Fy(0)34 b Fz(ou)h Fs(p)28 b Fy(=)f(0\))p Fs(:)83 3379 y Fz(On)36 b(dit)h(alors)g(que)f(la)h(suite)g Fs(a)1218 3394 y Fn(p)1258 3379 y Fs(a)1309 3394 y Fn(p)p Fo(\000)p Fm(1)1456 3379 y Fs(:)17 b(:)g(:)f(a)1638 3394 y Fm(0)1714 3379 y Fz(forme)36 b(l')n(\023)-47 b(ecritur)-5 b(e)36 b(de)h Fs(n)g Fz(en)f(b)-5 b(ase)37 b Fs(b)p Fz(.)g(Cette)81 3500 y(\023)-47 b(ecritur)-5 b(e)34 b(est)h(unique.)83 3685 y(D)n(\023)-47 b(emonstr)-5 b(ation.)193 b Fy({)48 b(D'ab)s(ord,)30 b(l'existence)j(de)e(l')m(\023)-46 b(ecriture)32 b(en)f(base)h Fs(b)f Fy(d'un)g(en)m(tier)327 3806 y(inf)m(\023)-46 b(erieur)45 b Fs(n)g Fy(strictemen)m(t)h(\022)-49 b(a)44 b Fs(b)h Fy(est)e(\023)-46 b(eviden)m(te)46 b(car)e(c)m(haque)j(nom)m (bre)e(inf)m(\023)-46 b(erieur)327 3926 y(strictemen)m(t)38 b(\022)-49 b(a)36 b Fs(b)g Fy(est)h(sa)g(propre)c(\023)-46 b(ecriture)37 b(en)g(base)g Fs(b)f Fy(:)g Fs(p)e Fy(=)g(0)h(et)i Fs(a)2899 3941 y Fm(0)2972 3926 y Fy(=)d Fs(n)p Fy(.)i(De)327 4046 y(m)m(^)-46 b(eme)34 b Fs(b)f Fy(s')m(\023)-46 b(ecrit)33 b(en)h(base)f Fs(b)g Fy(:)f(\\10")g(:)h Fs(p)27 b Fy(=)h(1)p Fs(;)17 b(a)2063 4061 y Fm(0)2130 4046 y Fy(=)27 b(0)p Fs(;)17 b(a)2377 4061 y Fm(1)2444 4046 y Fy(=)28 b(1.)327 4167 y(On)42 b(v)-5 b(a)42 b(mon)m(trer)h(main)m(tenan)m(t)g(mon)m (trer)g(l'existence)h(par)e(l'absurde.)i(Soit)e Fs(A)327 4287 y Fy(l'ensem)m(ble)35 b(des)f(en)m(tiers)g(qui)f(n'admetten)m(t)h (pas)f(d')m(\023)-46 b(ecriture)34 b(en)f(base)h Fs(b)p Fy(.)327 4407 y(On)43 b(supp)s(ose)h(par)f(l'absurde)h(que)h Fs(A)e Fy(est)g(non)h(vide.)g(Soit)f(donc)g Fs(n)g Fy(son)h(plus)327 4528 y(p)s(etit)32 b(\023)-46 b(el)m(\023)g(emen)m(t.)35 b(D'apr)m(\022)-46 b(es)35 b(ce)g(qui)f(pr)m(\023)-46 b(ec)m(\022)g(ede)36 b Fs(n)31 b(>)f(b)p Fy(.)35 b(E\013ectuons)h(main) m(tenan)m(t)f(la)327 4648 y(division)44 b(euclienne)g(de)f Fs(n)f Fy(par)g Fs(b)h Fy(:)f(on)e(\023)-46 b(ecrit)42 b Fs(n)j Fy(=)f Fs(bq)33 b Fy(+)28 b Fs(r)s Fy(,)42 b(a)m(v)m(ec)i Fs(q)i Fy(en)m(tier)e(et)327 4769 y(0)34 b Fp(\024)h Fs(r)i(<)e(b)p Fy(.)i(On)g(a)f(forc)m(\023)-46 b(emen)m(t)38 b Fs(q)g Fp(6)p Fy(=)c(0,)j(sinon)g(on)g(aurait)f Fs(n)f(<)f(b)p Fy(,)j(ce)g(qui)h(n'est)327 4889 y(pas)c(p)s(ossible.)i(On)e(ne)g(p)s (eut)h(pas)f(non)g(plus)h(a)m(v)m(oir)g Fs(q)f Fy(=)c(1,)k(car)g(alors) g(on)g(aurait)327 5009 y(on)d(a)f Fs(n)e Fy(=)g Fs(b)19 b Fy(+)g Fs(r)s Fy(,)31 b(ce)g(qui)h(constituerait)g(une)d(\023)-46 b(ecriture)32 b(de)f Fs(n)g Fy(en)h(base)f Fs(b)h Fy(:)f(\\1)p Fs(r)s Fy(")f(:)327 5130 y Fs(p)e Fy(=)f(1)p Fs(;)17 b(a)651 5145 y Fm(0)718 5130 y Fy(=)27 b Fs(r)m(;)17 b(a)957 5145 y Fm(1)1025 5130 y Fy(=)27 b(1.)33 b(Comme)g Fs(q)f(>)27 b Fy(1,)33 b(on)f(a)h(clairemen)m(t)h Fs(n)28 b Fp(\025)g Fy(2)p Fs(b)g(>)f(b)p Fy(.)327 5250 y(Comme)43 b Fs(b)g Fy(est)g(strictemen)m(t)h(plus)f(p)s(etit)f(que)h(le)g(plus)g (p)s(etit)f(des)h(en)m(tiers)h(qui)327 5370 y(n'admetten)m(t)36 b(pas)e(d')m(\023)-46 b(ecriture)36 b(en)e(base)h Fs(b)p Fy(,)g Fs(b)f Fy(admet)h(une)d(\023)-46 b(ecriture)35 b(en)f(base)h Fs(b)g Fy(:)p eop end %%Page: 10 14 TeXDict begin 10 13 bop 432 291 a @beginspecial @setspecial @endspecial Fy(10)1350 b Ft(CHAPITRE)34 b(1.)65 b(CONSTR)m(UCTION)35 b(DE)e Fu(N)676 555 y Fy(on)f(p)s(eut)e(\023)-46 b(ecrire)1922 737 y Fs(q)31 b Fy(=)2147 624 y Fn(p)2104 655 y Fk(X)2100 837 y Fn(k)r Fm(=0)2245 737 y Fs(a)2296 752 y Fn(k)2339 737 y Fs(b)2380 696 y Fn(k)676 956 y Fy(a)m(v)m(ec)1118 1081 y Fp(8)p Fs(k)g Fp(2)d(f)p Fy(0)p Fs(;)17 b(:)g(:)g(:)e(;)i(p)p Fp(g)97 b Fy(0)28 b Fp(\024)g Fs(a)2095 1096 y Fn(k)2165 1081 y Fs(<)g(b)33 b Fy(et)g(\()p Fs(a)2546 1096 y Fn(p)2613 1081 y Fs(>)28 b Fy(0)k(ou)h Fs(p)27 b Fy(=)h(0\))p Fs(:)676 1260 y Fy(Main)m(tenan)m(t)1595 1422 y Fs(n)f Fy(=)h Fs(r)d Fy(+)d Fs(bq)32 b Fy(=)27 b Fs(r)e Fy(+)2384 1310 y Fn(p)2342 1340 y Fk(X)2337 1523 y Fn(k)r Fm(=0)2482 1422 y Fs(a)2533 1437 y Fn(k)2576 1422 y Fs(b)2617 1381 y Fn(k)r Fm(+1)676 1668 y Fy(Ainsi)g Fs(a)971 1683 y Fn(p)1010 1668 y Fs(a)1061 1683 y Fn(p)p Fo(\000)p Fm(1)1208 1668 y Fs(:)17 b(:)g(:)f(a)1390 1683 y Fm(0)1430 1668 y Fs(r)26 b Fy(constitue)g(une)c(\023)-46 b(ecriture)24 b(en)h(base)g Fs(b)f Fy(de)h Fs(n)p Fy(,)f(d'o)s(\022)-51 b(u)23 b(la)h(con)m(tra-)676 1788 y(diction.)578 1909 y({)49 b(Mon)m(trons)24 b(d'ab)s(ord)f(que)h(si)f(en)h(en)m(tier)g (admet)g(deux)d(\023)-46 b(ecritures,)25 b(elles)f(admetten)m(t)676 2029 y(n)m(\023)-46 b(ecessairemen)m(t)36 b(le)d(m)m(^)-46 b(eme)33 b(nom)m(bre)h(de)f(c)m(hi\013res)h(:)f(si)g(on)f(a)1916 2291 y Fs(n)c Fy(=)2152 2178 y Fn(p)2110 2209 y Fk(X)2106 2392 y Fn(k)r Fm(=0)2251 2291 y Fs(a)2302 2306 y Fn(k)2345 2291 y Fs(b)2386 2250 y Fn(k)676 2555 y Fy(a)m(v)m(ec)1118 2680 y Fp(8)p Fs(k)j Fp(2)d(f)p Fy(0)p Fs(;)17 b(:)g(:)g(:)e(;)i(p)p Fp(g)97 b Fy(0)28 b Fp(\024)g Fs(a)2095 2695 y Fn(k)2165 2680 y Fs(<)g(b)33 b Fy(et)g(\()p Fs(a)2546 2695 y Fn(p)2613 2680 y Fs(>)28 b Fy(0)k(ou)h Fs(p)27 b Fy(=)h(0\))p Fs(:)676 2859 y Fy(Si)k Fs(p)c Fy(=)g(0,)k(on)g(a)h(0)27 b Fp(\024)h Fs(n)g(<)g(b)p Fy(.)33 b(Sinon,)g(on)f(a)1317 3140 y Fs(b)1358 3099 y Fn(p)1426 3140 y Fp(\024)c Fs(a)1582 3155 y Fn(p)1622 3140 y Fs(b)1663 3099 y Fn(p)1731 3140 y Fp(\024)g Fs(n)g Fp(\024)2074 3028 y Fn(p)2031 3058 y Fk(X)2027 3241 y Fn(k)r Fm(=0)2156 3140 y Fy(\()p Fs(b)22 b Fp(\000)h Fy(1\))p Fs(b)2485 3099 y Fn(k)2555 3140 y Fy(=)28 b Fs(b)2700 3099 y Fn(p)p Fm(+1)2852 3140 y Fp(\000)23 b Fy(1)p Fs(:)676 3431 y Fy(Ainsi,)29 b(si)f(un)h(en)m(tier) g(naturel)f(non)g(n)m(ul)h Fs(n)f Fy(admet)h(une)d(\023)-46 b(ecriture)29 b(\022)-49 b(a)28 b Fs(p)13 b Fy(+)g(1)27 b(c)m(hi\013res,)676 3551 y(on)i(a)h Fs(b)928 3515 y Fn(p)996 3551 y Fp(\024)e Fs(n)g Fp(\024)g Fs(b)1333 3515 y Fn(p)p Fm(+1)1480 3551 y Fp(\000)17 b Fy(1.)29 b(Donc)h(si)g Fs(n)g Fy(admet)h(une)c(\023)-46 b(ecriture)31 b(\022)-49 b(a)30 b Fs(p)17 b Fy(+)g(1)28 b(c)m(hi\013res)j(et)676 3672 y(une)25 b(\023)-46 b(ecriture)28 b(\022)-49 b(a)26 b Fs(q)15 b Fy(+)c(1)27 b(c)m(hi\013res,)i(on)e(a)f Fs(b)2102 3635 y Fn(p)2170 3672 y Fp(\024)i Fs(n)g 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b(de)f(la)f(division)i(de)f Fs(q)2802 4272 y Fn(k)2878 4257 y Fy(par)f Fs(b)676 4480 y Fy(et)g(d)m(\023)-46 b(e\014nissons)35 b(\()p Fs(q)1354 4495 y Fn(k)1397 4480 y Fy(\))1435 4495 y Fm(0)p Fo(\024)p Fn(k)r Fo(\024)p Fn(p)1691 4480 y Fy(par)d(la)h(form)m(ule)1386 4705 y Fs(r)1430 4720 y Fn(k)1500 4705 y Fy(=)60 b(reste)34 b(de)f(la)f(division)i(de)f Fs(q)2642 4720 y Fn(k)2717 4705 y Fy(par)g Fs(b:)676 4929 y Fy(De)h(la)h(sorte,)g(on)g(a)f Fs(q)1471 4944 y Fn(k)1545 4929 y Fy(=)d Fs(bq)1736 4944 y Fn(k)r Fm(+1)1894 4929 y Fy(+)23 b Fs(r)2037 4944 y Fn(k)2080 4929 y Fy(,)35 b(a)m(v)m(ec)h(0)31 b Fp(\024)h Fs(r)2591 4944 y Fn(k)2665 4929 y Fs(<)f(b)p Fy(.)k(Consid)m(\023)-46 b(erons)37 b(main-)676 5050 y(tenan)m(t)c(une)d(\023)-46 b(ecriture)34 b(de)f Fs(n)g Fy(en)g(base)g Fs(b)p Fy(.)1916 5311 y Fs(n)28 b Fy(=)2152 5199 y Fn(p)2110 5229 y Fk(X)2106 5412 y Fn(k)r Fm(=0)2251 5311 y Fs(a)2302 5326 y Fn(k)2345 5311 y Fs(b)2386 5270 y Fn(k)p eop end %%Page: 11 15 TeXDict begin 11 14 bop 83 291 a @beginspecial @setspecial @endspecial Ft(1.6.)65 b(EXER)m(CICES)2354 b Fy(11)327 555 y(a)m(v)m(ec)50 b Fp(8)p Fs(k)58 b Fp(2)e(f)p Fy(0)p Fs(;)17 b(:)g(:)g(:)e(;)i(p)p Fp(g)97 b Fy(0)55 b Fp(\024)g Fs(a)1643 570 y Fn(k)1741 555 y Fs(<)g(b:)49 b Fy(On)g(v)-5 b(a)48 b(mon)m(trer)h(que)h(p)s(our)e(tout)327 676 y Fs(i)f Fp(2)g(f)p Fy(0)p Fs(;)17 b(:)g(:)g(:)e(;)i(p)p Fp(g)p Fy(,)43 b(on)h(a)g Fs(r)1290 691 y Fn(i)1364 676 y Fy(=)j Fs(a)1538 691 y Fn(i)1566 676 y Fy(,)d(ce)g(qui)h(mon)m(trera) f(l'unicit)m(\023)-46 b(e)45 b(P)m(osons,)g(p)s(our)327 796 y(0)27 b Fp(\024)i Fs(i)e Fp(\024)i Fs(p)j Fy(:)1495 974 y Fs(Q)1572 989 y Fn(i)1628 974 y Fy(=)1774 861 y Fn(p)1732 892 y Fk(X)1733 1074 y Fn(k)r Fm(=)p Fn(i)1868 974 y Fs(a)1919 989 y Fn(k)1962 974 y Fs(b)2003 933 y Fn(k)r Fo(\000)p Fn(i)2125 974 y Fs(:)327 1215 y Fy(Il)38 b(est)g(ais)m(\023)-46 b(e)38 b(de)g(v)m(oir)g(que)h(l'on)e(a)h Fs(Q)1659 1230 y Fn(i)1723 1215 y Fy(=)e Fs(bQ)1953 1230 y Fn(i)p Fm(+1)2098 1215 y Fy(+)25 b Fs(a)2250 1230 y Fn(i)2279 1215 y Fy(,)37 b(de)h(telle)h(sorte)f(que)g Fs(Q)3202 1230 y Fn(i)p Fm(+1)327 1335 y Fy(est)f(quotien)m(t)g(de)f (la)g(division)i(de)e Fs(Q)1691 1350 y Fn(i)1756 1335 y Fy(par)g Fs(b)p Fy(,)g(et)h Fs(a)2206 1350 y Fn(i)2270 1335 y Fy(le)f(reste)h(de)g(la)f(division)h(de)327 1455 y Fs(Q)404 1470 y Fn(i)465 1455 y Fy(par)32 b Fs(b)p Fy(.)h(D'autre)g(part,)f(il)h(est)g(clair)g(que)g Fs(Q)2048 1470 y Fm(0)2116 1455 y Fy(=)27 b Fs(n)p Fy(.)33 b(Ainsi,)h(on)e(a)813 1593 y Fk(8)813 1679 y(>)813 1715 y(>)813 1750 y(<)813 1898 y(>)813 1934 y(>)813 1969 y(:)928 1668 y Fs(q)971 1683 y Fm(0)1038 1668 y Fy(=)c Fs(n)928 1788 y(Q)1005 1803 y Fm(0)1072 1788 y Fy(=)g Fs(n)928 1908 y(q)971 1923 y Fn(k)r Fm(+1)1132 1908 y Fy(=)60 b(quotien)m(t)34 b(de)f(la)f(division)i(de)f Fs(q)2425 1923 y Fn(k)2501 1908 y Fy(par)f Fs(b)928 2029 y(Q)1005 2044 y Fn(k)r Fm(+1)1166 2029 y Fy(=)60 b(quotien)m(t)34 b(de)f(la)f(division)i(de)f Fs(Q)2493 2044 y Fn(k)2569 2029 y Fy(par)f Fs(b)327 2243 y Fy(Ainsi,)46 b(il)f(est)h(ais)m(\023)-46 b(e)45 b(de)h(mon)m(trer)f (par)g(r)m(\023)-46 b(ecurrence)47 b(que)f Fs(q)2518 2258 y Fn(k)2610 2243 y Fy(=)i Fs(Q)2811 2258 y Fn(k)2899 2243 y Fy(p)s(our)d(tout)327 2363 y Fs(i)28 b Fp(2)g(f)p Fy(0)p Fs(;)17 b(:)g(:)g(:)e(;)i(p)p Fp(g)p Fy(.)327 2483 y(Comme)31 b Fs(a)733 2498 y Fn(i)791 2483 y Fy(le)f(reste)h(de)f (la)f(division)i(de)f Fs(Q)1916 2498 y Fn(i)1975 2483 y Fy(par)f Fs(b)h Fy(et)g Fs(r)2371 2498 y Fn(i)2429 2483 y Fy(le)g(reste)h(de)f(la)f(division)327 2604 y(de)k Fs(q)500 2619 y Fn(i)561 2604 y Fy(par)f Fs(b)p Fy(,)h(il)g(s'ensuit)h (que)g Fs(a)1508 2619 y Fn(i)1564 2604 y Fy(=)28 b Fs(r)1712 2619 y Fn(i)1772 2604 y Fy(p)s(our)33 b(tout)f Fs(i)c Fp(2)g(f)p Fy(0)p Fs(;)17 b(:)g(:)g(:)e(;)i(p)p Fp(g)p Fy(.)p 3250 2724 4 66 v 3254 2662 59 4 v 3254 2724 V 3312 2724 4 66 v 83 2875 a @beginspecial @setspecial @endspecial 183 x Fq(1.6)161 b(Exercices)83 3140 y @beginspecial @setspecial @endspecial 202 3278 a Fy(1.)49 b(P)m(our)25 b(c)m(hacun)h(des)g(couples)h(suiv)-5 b(an)m(ts)26 b(\()p Fs(X)8 b Fy(,application\))25 b(,)g(dites)h(si)f(l'application)327 3398 y(est)46 b(une)g(loi)f(de)g(comp)s(osition)h(in)m(terneou)g(une)g (loi)f(de)h(comp)s(osition)g(in)m(terne)327 3518 y(asso)s(ciativ)m (esur)35 b Fs(X)8 b Fy(.)327 3656 y(Dans)32 b(tous)h(les)h(cas,)f(on)f (donnera)h(une)h(preuv)m(e)g(du)f(r)m(\023)-46 b(esultat)33 b(annonc)m(\023)-46 b(e.)327 3777 y({)48 b Fp(\002)33 b Fy(sur)h Fu(R)770 3792 y Fm(+)829 3777 y Fy(.)327 3897 y({)48 b Fp(\002)33 b Fy(sur)h Fu(R)770 3912 y Fo(\000)829 3897 y Fy(.)327 4017 y({)48 b Fp(\000)33 b Fy(sur)h Fu(R)770 4032 y Fo(\000)829 4017 y Fy(.)327 4138 y({)48 b(\()p Fs(x;)17 b(y)t Fy(\))27 b Fp(7!)h Fs(x)22 b Fp(^)h Fs(y)31 b Fy(=)c(max)q(\()p Fs(x;)17 b(y)t Fy(\))31 b(sur)i Fu(R)p Fy(.)327 4258 y({)48 b(\()p Fs(x;)17 b(y)t Fy(\))27 b Fp(7!)h Fs(x)22 b(?)g(y)31 b Fy(=)c Fp(j)p Fs(x)c Fp(\000)f Fs(y)t Fp(j)32 b Fy(sur)h Fu(R)p Fy(.)327 4378 y({)48 b(\()p Fs(x;)17 b(y)t Fy(\))27 b Fp(7!)h Fs(x)22 b(?)g(y)31 b Fy(=)c Fp(j)p Fs(x)c Fp(\000)f Fs(y)t Fp(j)32 b Fy(sur)h Fp(f)p Fy(0)p Fs(;)17 b Fy(1)p Fp(g)p Fy(.)83 4421 y @beginspecial @setspecial @endspecial 202 4534 a(2.)49 b(Dans)38 b(un)h(mono)-11 b(\177)-38 b(\020de)39 b(quelconque,)j(mon)m (trez)d(que)h(si)f Fs(a)26 b(?)g(x)39 b Fy(=)e Fs(e)i Fy(et)g Fs(x)26 b(?)g(b)39 b Fy(=)f Fs(e)p Fy(,)327 4654 y(alors)32 b(n)m(\023)-46 b(ecessairemen)m(t)36 b Fs(a)28 b Fy(=)g Fs(b)83 4671 y @beginspecial @setspecial @endspecial 202 4809 a Fy(3.)49 b(Si)37 b Fs(A)g Fy(et)g Fs(B)43 b Fy(son)m(t)37 b(deux)h(parties)g(d'un)g(ensem)m(ble)h Fs(E)6 b Fy(,)37 b(on)g(d)m(\023)-46 b(e\014nit)38 b(la)f(di\013)m (\023)-46 b(erence)327 4929 y(sym)m(\023)g(etrique)35 b(de)e Fs(A)g Fy(et)g Fs(B)k Fy(:)1264 5150 y Fs(A)p Fy(\001)p Fs(B)c Fy(=)28 b(\()p Fs(A)22 b Fp([)h Fs(B)5 b Fy(\))p Fp(n)p Fy(\()p Fs(A)22 b Fp(\\)g Fs(B)5 b Fy(\))p Fs(:)336 5345 y Fy(\023)327 5370 y(Etudier)33 b(les)h(propri)m(\023)-46 b(et)m(\023)g(e)33 b(de)g(la)g(loi)f(\001)h(sur)g Fp(P)8 b Fy(\()p Fs(E)e Fy(\).)p eop end %%Page: 12 16 TeXDict begin 12 15 bop 432 291 a @beginspecial @setspecial @endspecial Fy(12)1350 b Ft(CHAPITRE)34 b(1.)65 b(CONSTR)m(UCTION)35 b(DE)e Fu(N)432 456 y @beginspecial @setspecial @endspecial 551 555 a Fy(4.)49 b(Soien)m(t)27 b Fs(x)g Fy(et)g Fs(y)i Fy(deux)c(\023)-46 b(el)m(\023)g(emen)m(ts)29 b(in)m(v)m(ersibles)g (d'un)f(mono)-11 b(\177)-38 b(\020de.)27 b(Mon)m(trer)g(que)h Fs(x)10 b(?)g(y)676 676 y Fy(est)35 b(in)m(v)m(ersible.)j(R)m(\023)-46 b(ecipro)s(quemen)m(t,)37 b(si)f Fs(x)24 b(?)f(y)38 b Fy(est)e(in)m(v)m(ersible,)h(p)s(eut)f(on)e(a\016c)m(her)676 796 y(que)f Fs(x)g Fy(et)g Fs(y)j Fy(son)m(t)d(in)m(v)m(ersibles)16 b(?)432 832 y @beginspecial @setspecial @endspecial 551 951 a(5.)49 b(Si)34 b Fs(u)f Fy(et)h Fs(v)k Fy(son)m(t)32 b(\023)-46 b(el)m(\023)g(emen)m(ts)36 b(de)f(])23 b Fp(\000)g Fy(1)p Fs(;)17 b Fy(1[,)34 b(on)g(d)m(\023)-46 b(e\014nit)35 b(:)f Fs(u)22 b Fp(\003)h Fs(v)34 b Fy(=)3071 911 y Fn(u)p Fm(+)p Fn(v)p 3054 927 168 4 v 3054 985 a Fm(1+)p Fn(uv)3232 951 y Fy(.)g Fp(\003)f Fy(est-elle)676 1071 y(une)g(loi)f(de)h(group)s (e)g(sur)g(])22 b Fp(\000)h Fy(1)p Fs(;)17 b Fy(1[)f(?)432 1113 y @beginspecial @setspecial @endspecial 551 1226 a(6.)49 b(Soien)m(t)31 b Fs(a;)17 b(b)31 b Fy(deux)e(\023)-46 b(el)m(\023)g(emen)m(ts)32 b(d'un)f(group)g(\()p Fs(G;)17 b(?)p Fy(\))30 b(d')m(\023)-46 b(el)m(\023)g(emen)m(t)32 b(neutre)g Fs(cp=v)t(idee)p Fy(,)676 1346 y Fs(n)27 b Fy(un)g(en)m(tier)h(naturel.)f(On)g(supp)s(ose)h(que)g(\()p Fs(a)10 b(?)g(b)p Fy(\))2452 1310 y Fn(n)2528 1346 y Fy(=)27 b Fs(e)p Fy(.)g(Mon)m(trer)h(que)g(\()p Fs(b)10 b Fp(\003)g Fs(a)p Fy(\))3518 1310 y Fn(n)3594 1346 y Fy(=)676 1467 y Fs(e)p Fy(.)432 1484 y @beginspecial @setspecial @endspecial 551 1621 a(7.)49 b(Mon)m(trer)22 b(qu'il)h(n'existe)g(pas)g(de)f(suite)g(in\014nie)h(\()p Fs(u)2478 1636 y Fn(n)2525 1621 y Fy(\))2563 1636 y Fn(n)p Fo(\025)p Fm(0)2721 1621 y Fy(strictemen)m(t)h(d)m(\023)-46 b(ecroissan)m(te)676 1742 y(\022)d(a)32 b(v)-5 b(aleurs)33 b(dans)g Fu(N)p Fy(.)432 1759 y @beginspecial @setspecial @endspecial 551 1896 a(8.)684 1871 y(\023)676 1896 y(Ecrire)g (12,57,128)e(en)i(base)h(3.)432 1933 y @beginspecial @setspecial @endspecial 551 2051 a(9.)49 b(Commen)m(t)27 b(s')m(\023)-46 b(ecrit)27 b(en)f(base)g(7)g(la)f(somme)i(du)f(nom)m (bre)h(qui)f(s')m(\023)-46 b(ecrit)27 b(1286)e(en)h(base)676 2171 y(7)32 b(et)h(du)g(nom)m(bre)g(qui)g(s')m(\023)-46 b(ecrit)34 b(3126)e(en)h(base)g(7.)432 2208 y @beginspecial @setspecial @endspecial 502 2326 a(10.)49 b(Un)37 b(en)m(tier)i (naturel)f Fs(n)g Fy(est)g(dit)f(pratique)i(si)f(c)m(haque)h(en)m(tier) g(naturel)f(inf)m(\023)-46 b(erieur)676 2446 y(ou)25 b(\023)-46 b(egal)29 b(\022)-49 b(a)28 b Fs(n)h Fy(p)s(eut)g(s')m(\023) -46 b(ecrire)30 b(comme)g(somme)g(de)f(diviseurs)i(distincts)f(de)f Fs(n)p Fy(.)g(P)m(ar)676 2567 y(exemple)i(6)e(est)g(pratique,)h(car)g (1)d(=)h(1)p Fs(;)17 b Fy(2)26 b(=)i(2)p Fs(;)17 b Fy(3)27 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Fs(:)432 4935 y @beginspecial @setspecial @endspecial 130 x Fw(2.1.2)136 b(Ensem)l(ble)46 b(quotien)l(t)432 5250 y Fx(D)n(\023)-54 b(e\014nition:)46 b Fy(Soit)39 b Fp(R)h Fy(une)g(relation)f(d')m(\023)-46 b(equiv)-5 b(alence)42 b(sur)e Fs(X)8 b Fy(.)39 b(On)g(app)s(elle)h (quotien)m(t)432 5370 y(de)29 b Fs(X)37 b Fy(par)29 b Fp(R)h Fy(et)f(on)g(note)g Fs(X=)p Fp(R)g Fy(l'ensem)m(ble)j(des)e (classes)h(d')m(\023)-46 b(equiv)-5 b(alences)32 b(de)e Fs(X)37 b Fy(p)s(our)p eop end %%Page: 15 19 TeXDict begin 15 18 bop 83 291 a @beginspecial @setspecial @endspecial Ft(2.1.)65 b(RELA)-8 b(TIONS)33 b(D')1020 265 y(\023)1007 291 y(EQUIV)-11 b(ALENTES)1441 b Fy(15)83 555 y(la)32 b(relation)h Fp(R)p Fy(.)83 676 y Fx(Exemple:)k Fy(P)m(our)32 b(la)f(relation)g(d')m(\023)-46 b(equiv)-5 b(alence)34 b(sur)e Fs(X)j Fy(=)28 b Fp(f\000)p Fy(2)p Fs(;)17 b Fp(\000)p Fy(1)p Fs(;)g Fy(0)p Fs(;)g Fy(1)p Fs(;)g Fy(2)p Fs(;)g Fy(3)p Fp(g)28 b Fy(d)m(\023)-46 b(e\014nie)83 796 y(par)32 b Fs(x)p Fp(R)p Fs(y)60 b Fp(\()-17 b(\))55 b Fy(\()p Fp(j)p Fs(x)p Fp(j)27 b Fy(=)h Fp(j)p Fs(y)t Fp(j)p Fy(\),)j(on)h(a)h Fs(X=)p Fp(R)27 b Fy(=)h Fp(ff\000)p Fy(2;)17 b(2)p Fp(g)p Fs(;)g Fp(f\000)p Fy(1;)g(1)p Fp(g)p Fs(;)g Fp(f)p Fy(0)p Fp(g)p Fs(;)g Fp(f)p Fy(3)p Fp(gg)p Fy(.)229 916 y(Remarque)32 b(:)e(sur)h(un)g (ensemle)h Fs(X)38 b Fy(quelconque,)33 b(l')m(\023)-46 b(egalit)m(\023)g(e)31 b(est)g(toujours)f(une)h(rela-)83 1037 y(tion)23 b(d')m(\023)-46 b(equiv)-5 b(alence.)25 b(Dans)e(ce)h(cas,)f(c)m(haque)i(classe)f(d')m(\023)-46 b(equiv)-5 b(alence)25 b(est)f(constitu)m(\023)-46 b(e)24 b(d'un)83 1157 y(unique)38 b(\023)-46 b(el)m(\023)g(emen)m(t.)40 b(Et)f(on)g(a)g Fs(X=)p Fp(R)f Fy(=)g Fp(ff)p Fs(x)p Fp(g)p Fy(;)17 b Fs(x)39 b Fp(2)f Fs(X)8 b Fp(g)p Fy(,)39 b(que)h(l'on)f(p)s(eut)d(\023)-46 b(evidemmen)m(t)83 1277 y(iden)m(ti\014er)34 b(\022)-49 b(a)32 b Fs(X)8 b Fy(.)-606 b @beginspecial @setspecial @endspecial 186 x Fx(Th)n(\023)-54 b(eor)n(\022)g(eme)36 b(7.)k Fz(Soit)33 b Fp(R)h Fz(une)f(r)-5 b(elation)33 b(d')n(\023)-47 b(equivalenc)-5 b(e)30 b(sur)k(un)f(ensemble)f Fs(X)8 b Fz(,)32 b Fp(R)i Fz(une)83 1584 y(r)-5 b(elation)41 b(d')n(\023)-47 b(equivalenc)-5 b(e)40 b(sur)i(un)g(ensemble)e Fs(X)1878 1547 y Fo(0)1943 1584 y Fz(et)i Fs(f)53 b Fz(une)42 b(applic)-5 b(ation)40 b(de)i Fs(X)50 b Fz(dans)83 1704 y Fs(X)172 1668 y Fo(0)195 1704 y Fz(.)229 1824 y(On)35 b(supp)-5 b(ose)34 b(que)928 2024 y Fp(8)p Fy(\()p Fs(x;)17 b(y)t Fy(\))28 b Fp(2)g Fs(X)1421 1983 y Fm(2)1560 2024 y Fs(x)p Fp(R)p Fs(y)j Fy(=)-17 b Fp(\))28 b Fs(f)11 b Fy(\()p Fs(x)p Fy(\))p Fp(R)2239 1983 y Fo(0)2263 2024 y Fs(f)g Fy(\()p Fs(y)t Fy(\))p Fs(:)83 2223 y Fz(A)n(lors,)34 b(il)h(existe)g(une)f(applic)-5 b(ation)p 1417 2144 59 4 v 34 w Fs(f)46 b Fz(de)34 b Fs(X=)p Fp(R)h Fz(dans)f Fs(X)2207 2187 y Fo(0)2230 2223 y Fs(=)p Fp(R)2363 2187 y Fo(0)2421 2223 y Fz(tel)5 b(le)35 b(que)1222 2422 y Fp(8)p Fs(x)29 b Fp(2)f Fs(X)p 1643 2338 191 4 v 107 w(f)11 b Fy(\()p Fs(x)p Fy(\))28 b(=)p 1964 2343 59 4 v 27 w Fs(f)11 b Fy(\()p 2061 2370 56 4 v Fs(x)q Fy(\))p Fs(:)83 2622 y Fz(On)34 b(dit)h(alors)g(que)g (l'applic)-5 b(ation)33 b Fs(f)46 b Fz(p)-5 b(asse)34 b(au)h(quotient.)83 2808 y(D)n(\023)-47 b(emonstr)-5 b(ation.)46 b Fy(Soit)e Fs(C)54 b Fp(2)48 b Fs(X=)p Fp(R)p Fy(.)c Fs(C)51 b Fy(est)44 b(une)h(partie)f(non)g(vide)h(de)g Fs(X)8 b Fy(.)44 b(Notons)83 2928 y Fs(A)30 b Fy(=)h Fs(f)11 b Fy(\()p Fs(C)c Fy(\).)33 b Fs(A)h Fy(est)h(non)g(vide)g(car)f Fs(C)41 b Fy(non)34 b(vide.)h(Soien)m(t)g Fs(y)2276 2943 y Fm(1)2349 2928 y Fy(et)g Fs(y)2513 2943 y Fm(2)2586 2928 y Fy(deux)d(\023)-46 b(el)m(\023)g(emen)m(ts)36 b(de)83 3048 y Fs(A)p Fy(.)j(P)m(ar)g(d)m(\023)-46 b(e\014nition)40 b(de)f Fs(A)p Fy(,)g(il)g(existe)h Fs(x)1554 3063 y Fm(1)1633 3048 y Fy(et)f Fs(x)1808 3063 y Fm(2)1886 3048 y Fy(dans)h Fs(C)45 b Fp(\032)38 b Fs(X)47 b Fy(tels)39 b(que)h Fs(y)2900 3063 y Fm(1)2977 3048 y Fy(=)e Fs(f)11 b Fy(\()p Fs(x)3243 3063 y Fm(1)3283 3048 y Fy(\))83 3169 y(et)34 b Fs(y)246 3184 y Fm(2)313 3169 y Fy(=)29 b Fs(f)11 b Fy(\()p Fs(x)570 3184 y Fm(2)610 3169 y Fy(\).)33 b(P)m(ar)g(d)m(\023)-46 b(e\014nition)35 b(d'une)f(classe)h(d')m(\023)-46 b(equiv)-5 b(alence,)35 b(on)f(a)f Fs(x)2762 3184 y Fm(1)2802 3169 y Fp(R)p Fs(x)2941 3184 y Fm(2)2981 3169 y Fy(,)g(ce)h(qui,)83 3289 y(par)43 b(h)m(yp)s(oth)m(\022)-46 b(ese,)46 b(en)m(tra)-11 b(^)-38 b(\020ne)44 b Fs(y)1199 3304 y Fm(1)1238 3289 y Fp(R)p Fs(y)1370 3304 y Fm(2)1410 3289 y Fy(,)f(soit)p 1676 3236 88 4 v 44 w Fs(y)1724 3304 y Fm(1)1809 3289 y Fy(=)p 1931 3236 V 46 w Fs(y)1979 3304 y Fm(2)2018 3289 y Fy(.)h(Ainsi)p 2352 3211 74 4 v 44 w Fs(A)g Fy(est)g(un)g (singleton,)g(ce)83 3409 y(qui)38 b(signi\014e)h(que)f(p)s(our)f(tout)g Fs(C)43 b Fp(2)37 b Fs(X=)p Fp(R)p Fy(,)g(il)g(existe)i(un)f(unique)h Fs(D)g Fp(2)d Fs(X=)p Fp(R)h Fy(tel)h(que)83 3530 y Fs(f)11 b Fy(\()p Fs(C)c Fy(\))27 b Fp(\032)h Fs(D)s Fy(.)33 b(On)f(v)-5 b(a)33 b(donc)g(d)m(\023)-46 b(e\014nir)p 1397 3451 59 4 v 33 w Fs(f)11 b Fy(\()p Fs(C)c Fy(\))32 b(par)p 1815 3451 V 33 w Fs(f)10 b Fy(\()p Fs(C)d Fy(\))28 b(=)f Fs(D)s Fy(.)229 3650 y(V)m(\023)-46 b(eri\014ons)29 b(main)m(tenan)m(t)g(que)f(l'application)h(d)m(\023)-46 b(e\014nie)28 b(satisfait)g(bien)h(\022)-49 b(a)27 b(la)g(condition)83 3771 y(v)m(oulue.)36 b(Soit)e Fs(x)c Fp(2)h Fs(X)24 b Fy(;)34 b(notons)h Fs(C)i Fy(=)p 1501 3718 56 4 v 30 w Fs(x)e Fy(et)f Fs(D)f Fy(=)p 1926 3691 59 4 v 30 w Fs(f)11 b Fy(\()p Fs(C)c Fy(\))34 b(:)p 2233 3686 212 4 v 34 w Fs(f)11 b Fy(\()p Fs(C)c Fy(\))30 b Fp(\032)g Fs(D)s Fy(.)k(En)h(particulier)83 3891 y Fs(x)28 b Fp(2)g Fs(C)7 b Fy(,)33 b(donc)g Fs(f)11 b Fy(\()p Fs(x)p Fy(\))28 b Fp(2)p 942 3812 59 4 v 28 w Fs(f)10 b Fy(\()p 1038 3838 56 4 v Fs(x)q Fy(\),)32 b(d'o)s(\022)-51 b(u)p 1408 3806 191 4 v 32 w Fs(f)11 b Fy(\()p Fs(x)p Fy(\))28 b(=)p 1729 3812 59 4 v 27 w Fs(f)11 b Fy(\()p 1826 3838 56 4 v Fs(x)p Fy(\))p Fs(:)p 3250 3891 4 66 v 3254 3828 59 4 v 3254 3891 V 3312 3891 4 66 v 83 4080 a Fx(Exemple:)30 b Fy(On)c(consid)m(\022)-46 b(ere)27 b(la)f(relation)g(d')m(\023)-46 b(equiv)-5 b(alence)28 b(sur)e Fs(X)36 b Fy(=)27 b Fp(f\000)p Fy(2)p Fs(;)17 b Fp(\000)p Fy(1)p Fs(;)g Fy(0)p Fs(;)g Fy(1)p Fs(;)g Fy(2)p Fs(;)g Fy(3)p Fp(g)83 4201 y Fy(d)m(\023)-46 b(e\014nie)36 b(par)e Fs(x)p Fp(R)p Fs(y)66 b Fp(\()-17 b(\))61 b Fy(\()p Fp(j)p Fs(x)p Fp(j)30 b Fy(=)h Fp(j)p Fs(y)t Fp(j)p Fy(\),)i(on)h(a)g Fs(X=)p Fp(R)d Fy(=)g Fp(ff\000)p Fy(2;)17 b(2)p Fp(g)p Fs(;)g Fp(f\000)p Fy(1;)g(1)p Fp(g)p Fs(;)g Fp(f)p Fy(0)p Fp(g)p Fs(;)g Fp(f)p Fy(3)p Fp(gg)p Fy(.)83 4321 y(Consid)m(\023)-46 b(erons)34 b(l'appication)1384 4520 y Fs(X)91 b Fp(!)82 b Fs(Y)49 b Fy(=)28 b Fu(N)1417 4666 y Fs(x)84 b Fp(7!)e Fs(x)1793 4624 y Fm(2)83 4865 y Fy(On)33 b(v)m(oit)g(que)g Fs(f)43 b Fy(passe)34 b(au)f(quotien)m(t.) 83 4884 y @beginspecial @setspecial @endspecial 167 x Fx(Corollaire)42 b(2.)i Fz(Soit)38 b Fs(X)46 b Fz(un)39 b(ensemble,)e(et)h Fs(?)g Fz(une)h(loi)f(le)g(c)-5 b(omp)g(osition)37 b(interne)h(sur)83 5171 y Fs(X)8 b Fz(.)35 b(On)f(supp)-5 b(ose)34 b(que)h Fp(\030)g Fz(est)g(une)g(r)-5 b(elation)34 b(d')n(\023)-47 b(equivalenc)-5 b(e)33 b(sur)i Fs(X)43 b Fz(tel)5 b(le)34 b(que)520 5370 y Fp(8)p Fy(\()p Fs(x;)17 b(x)767 5329 y Fo(0)791 5370 y Fs(;)g(y)t(;)g(y)983 5329 y Fo(0)1004 5370 y Fy(\))28 b Fp(2)g Fs(X)1253 5329 y Fm(4)1392 5370 y Fs(x)g Fp(\030)g Fs(x)1635 5329 y Fo(0)1693 5370 y Fz(et)36 b Fs(y)30 b Fp(\030)f Fs(y)2042 5329 y Fo(0)2092 5370 y Fy(=)-17 b Fp(\))27 b Fs(x)c(?)f(y)30 b Fp(\030)f Fs(x)2666 5329 y Fo(0)2712 5370 y Fs(?)21 b(y)2834 5329 y Fo(0)2857 5370 y Fs(;)p eop end %%Page: 16 20 TeXDict begin 16 19 bop 432 291 a @beginspecial @setspecial @endspecial Fy(16)2373 b Ft(CHAPITRE)34 b(2.)65 b Fu(Z)432 555 y Fz(alors)34 b(il)h(existe)f(une)h(loi)f(de)h(c)-5 b(omp)g(osition)34 b(interne)p 2362 499 49 4 v 34 w Fs(?)h Fz(sur)g Fs(X=)27 b Fp(\030)p Fz(tel)5 b(le)35 b(que)1470 772 y Fp(8)p Fy(\()p Fs(x;)17 b(y)t Fy(\))27 b Fp(2)h Fs(X)1962 731 y Fm(2)p 2101 716 200 4 v 2101 772 a Fs(x)23 b(?)e(y)31 b Fy(=)p 2432 719 56 4 v 28 w Fs(x)22 b(?)p 2580 719 52 4 v 22 w(y)432 989 y Fz(D)n(\023)-47 b(emonstr)-5 b(ation.)46 b Fy(On)33 b(d)m(\023)-46 b(e\014nit)33 b(une)g(relation)g (d')m(\023)-46 b(equiv)-5 b(alence)35 b Fp(R)e Fy(sur)g Fs(X)d Fp(\002)23 b Fs(X)40 b Fy(par)627 1206 y Fp(8)p Fy(\(\()p Fs(x;)17 b(y)t Fy(\))p Fs(;)g Fy(\()p Fs(x)1084 1165 y Fo(0)1107 1206 y Fs(;)g(y)1203 1165 y Fo(0)1225 1206 y Fy(\)\))28 b Fp(2)g Fs(X)1512 1165 y Fm(2)1573 1206 y Fp(\002)23 b Fs(X)1762 1165 y Fm(2)1899 1206 y Fy(\()p Fs(x;)17 b(y)t Fy(\))p Fp(R)p Fy(\()p Fs(x)2303 1165 y Fo(0)2326 1206 y Fs(;)g(y)2422 1165 y Fo(0)2444 1206 y Fy(\))55 b Fp(\()-17 b(\))55 b Fs(x)28 b Fp(\030)g Fs(x)3018 1165 y Fo(0)3075 1206 y Fy(et)33 b Fs(y)d Fp(\030)f Fs(y)3425 1165 y Fo(0)3447 1206 y Fs(:)432 1424 y Fy(On)35 b(d)m(\023)-46 b(e\014nit)37 b(alors)f Fs(f)46 b Fy(sur)36 b Fs(X)1490 1387 y Fm(2)1565 1424 y Fy(par)f Fs(f)11 b Fy(\(\()p Fs(x;)17 b(y)t Fy(\)\))32 b(=)h Fs(x)25 b(?)f(y)38 b Fy(est)f(on)e(applique)i(le)f(th)m(\023)-46 b(eor)m(\022)g(eme)432 1544 y(pr)m(\023)g(ec)m(\023)g(eden)m(t.)p 3599 1544 4 66 v 3603 1481 59 4 v 3603 1544 V 3661 1544 4 66 v 432 1713 a @beginspecial @setspecial @endspecial 163 x Fq(2.2)160 b(Construire)51 b Fl(Z)578 2095 y Fy(La)25 b(construction)i(de)f Fu(Z)g Fy(ne)g(v)-5 b(a)25 b(pas)h(de)g(soi.)g (L'id)m(\023)-46 b(ee)27 b(m)m(^)-46 b(eme)27 b(de)f(consid)m(\023)-46 b(erer)27 b(nom)m(bres)432 2216 y(n)m(\023)-46 b(egatifs)33 b(a)g(longtemps)f(\023)-46 b(et)m(\023)g(e)33 b(consid)m(\023)-46 b(er)m(\023)g(ee,)35 b(au)e(mieux)i(comme)f(une)g(astuce)g(d')m(\023) -46 b(ecriture,)432 2336 y(au)32 b(pis)h(comme)h(une)f(diablerie.)578 2456 y(Sans)22 b(un)g(cours)h(de)f(math)m(\023)-46 b(ematique)23 b(de)g(coll)m(\022)-46 b(ege,)22 b(on)g(in)m(tro)s(duit)g(souv)m(en)m (t)i(les)e(nom)m(bres)432 2577 y(relatifs)32 b(comme)h(repr)m(\023)-46 b(esen)m(tan)m(ts)35 b(d'un)d(d)m(\023)-46 b(eplacemen)m(t,)35 b(par)d(exemple)i(le)e(d)m(\023)-46 b(eplacemen)m(t)432 2697 y(d'un)42 b(ascenceur.)i(Ainsi)f(\\+2")e(est)i(ce)f(que)h(l'on)f (fait)g(p)s(our)g(fasser)g(du)h(premier)g(au)432 2818 y(troisi)m(\022)-46 b(eme)43 b(\023)-46 b(etage,)45 b(et)g(\\-2")f(ce)h (que)h(l'on)f(fait)g(p)s(our)f(passer)i(du)g(troisi)m(\022)-46 b(eme)46 b(au)f(pre-)432 2938 y(mier.)34 b(Le)g(probl)m(\022)-46 b(eme)33 b(\023)-46 b(etan)m(t)33 b(qu'il)i(ne)f(su\016t)h(pas,)f(p)s (our)g(d)m(\023)-46 b(e\014nir)35 b Fp(\000)p Fy(2)f(e\016cacemen)m(t,) i(de)432 3058 y(d)m(\023)-46 b(e\014nir)32 b(\\-2")e(comme)g(\023)-46 b(etan)m(t)32 b(ce)g(que)g(l'on)g(fait)f(p)s(our)g(passer)i(du)f(4e)c (\023)-46 b(etage)32 b(au)f(second,)432 3179 y(car)43 b(ce)h(faisan)m(t,)g(on)f(n)m(\023)-46 b(eglige,)45 b(ce)f(qui)g(est)g (tout)f(aussi)i(vrai,)e(que)i(\\-2")d(ce)i(que)g(l'on)432 3299 y(fait)32 b(p)s(our)g(passer)i(du)e(4e)e(\023)-46 b(etage)33 b(au)f(second.)i(P)m(our)f(p)s(ermettre)g(d'iden)m(ti\014er) i(ces)e(deux)432 3419 y(d)m(\023)-46 b(eplacemen)m(ts)34 b(comme)e(deux)g(a)m(v)-5 b(atars)31 b(d'un)h(m)m(^)-46 b(eme)33 b(ob)5 b(jet,)31 b(on)g(v)-5 b(a)31 b(utiliser)i(la)e(notion) 432 3540 y(d'ensem)m(ble)k(quotien)m(t.)432 3660 y Fx(D)n(\023)-54 b(e\014nition:)38 b Fy(Soit)32 b Fp(\030)h Fy(la)g(relation)g(d')m (\023)-46 b(equiv)-5 b(alence)35 b(sur)e Fu(N)22 b Fp(\002)h Fu(N)32 b Fy(d)m(\023)-46 b(e\014nie)34 b(par)1347 3877 y(\()p Fs(a;)17 b(b)p Fy(\))28 b Fp(\030)g Fy(\()p Fs(c;)17 b(d)p Fy(\))55 b Fp(\()-17 b(\))55 b Fs(a)22 b Fy(+)g Fs(d)27 b Fy(=)h Fs(b)22 b Fy(+)g Fs(c)432 4094 y Fy(On)k(p)s(ose)h Fu(Z)i Fy(=)e Fu(N)10 b Fp(\002)g Fu(N)p Fs(=)28 b Fp(\030)p Fy(.)f(Si)g(l'on)g(note)f(+)h(la)f(loi)h(de)g(comp)s(osition)g(in)m (terne)h(sur)f Fu(N)10 b Fp(\002)g Fu(N)432 4215 y Fy(d)m(\023)-46 b(e\014nie)34 b(par)1420 4335 y(\()p Fs(x;)17 b(y)t Fy(\))22 b(+)g(\()p Fs(z)t(;)17 b(t)p Fy(\))28 b(=)f(\()p Fs(x)c Fy(+)f Fs(z)t(;)17 b(y)26 b Fy(+)c Fs(t)p Fy(\))p Fs(:)432 4508 y Fy(Il)33 b(est)g(clair)g(que)g(que)596 4725 y(\()p Fs(x;)17 b(y)t Fy(\))27 b Fp(\030)h Fy(\()p Fs(x)1048 4684 y Fo(0)1072 4725 y Fs(;)17 b(y)1168 4684 y Fo(0)1190 4725 y Fy(\))33 b(et)f(\()p Fs(z)t(;)17 b(t)p Fy(\))29 b Fp(\030)f Fy(\()p Fs(z)1799 4684 y Fo(0)1823 4725 y Fs(;)17 b(t)1902 4684 y Fo(0)1925 4725 y Fy(\))28 b(=)-17 b Fp(\))27 b Fy(\()p Fs(x;)17 b(y)t Fy(\))k(+)h(\()p Fs(z)t(;)17 b(t)p Fy(\))29 b Fp(\030)f Fy(\()p Fs(x)2954 4684 y Fo(0)2978 4725 y Fs(;)17 b(y)3074 4684 y Fo(0)3096 4725 y Fy(\))22 b(+)g(\()p Fs(z)3341 4684 y Fo(0)3365 4725 y Fs(;)17 b(t)3444 4684 y Fo(0)3467 4725 y Fy(\))432 4942 y(Ceci)33 b(p)s(ermet)h(ainsi)f(de)g(d)m(\023)-46 b(e\014nir)34 b(une)f(addition)g(sur)g Fu(Z)g Fy(gr^)-49 b(ace)32 b(au)h(corollaire)g @beginspecial @setspecial @endspecial(2)p @beginspecial @setspecial @endspecial(.)578 5063 y(Quelques)i(propri)m(\023)-46 b(et)m(\023)g(es)33 b(simples)i(\(\022)-49 b(a)32 b(d)m(\023)-46 b(emon)m(trer)34 b(en)f(exercice\))432 5104 y @beginspecial @setspecial @endspecial 551 5217 a(1.)49 b Fp(8)p Fs(z)32 b Fp(2)c Fu(Z)p Fp(9)p Fy(!)p Fs(n)h Fp(2)f Fu(N)98 b Fs(z)32 b Fy(=)p 1582 5132 227 4 v 28 w(\()p Fs(n;)17 b Fy(0\))32 b(ou)g Fs(z)h Fy(=)p 2157 5132 V 27 w(\(0)p Fs(;)17 b(n)p Fy(\).)432 5258 y @beginspecial @setspecial @endspecial 551 5370 a(2.)49 b(Les)33 b(applications)g Fs(n)28 b Fp(7!)p 1607 5286 V 28 w Fy(\()p Fs(n;)17 b Fy(0\))32 b(et)g Fs(n)c Fp(7!)p 2193 5286 V 28 w Fy(\(0)p Fs(;)17 b(n)p Fy(\))32 b(son)m(t)h(injectiv)m(es.)p eop end %%Page: 17 21 TeXDict begin 17 20 bop 83 291 a @beginspecial @setspecial @endspecial Ft(2.3.)65 b(LA)32 b(STR)m(UCTURE)j(D'ANNEA)m(U)1575 b Fy(17)83 456 y @beginspecial @setspecial @endspecial 202 555 a(3.)49 b Fp(8)p Fy(\()p Fs(a;)17 b(b)p Fy(\))31 b Fp(2)f Fu(N)23 b Fp(\002)h Fu(N)p 1086 471 213 4 v 97 w Fy(\()p Fs(a;)17 b(b)p Fy(\))23 b(+)p 1420 471 V 23 w(\()p Fs(b;)17 b(a)p Fy(\))31 b(=)p 1769 471 218 4 v 30 w(\(0)p Fs(;)17 b Fy(0\))33 b(item)h Fp(8)p Fy(\()p Fs(a;)17 b(b)p Fy(\))31 b Fp(2)f Fu(N)23 b Fp(\002)h Fu(N)p 3002 471 220 4 v 97 w Fy(\()p Fs(a;)17 b Fy(0\))23 b(+)p 327 591 210 4 v 327 676 a(\()p Fs(b;)17 b Fy(0\))27 b(=)p 668 591 382 4 v 28 w(\()p Fs(a)22 b Fy(+)g Fs(b;)17 b Fy(0\))83 723 y @beginspecial @setspecial @endspecial 202 841 a(4.)49 b Fp(8)p Fs(z)33 b Fp(2)28 b Fu(Z)98 b Fs(z)27 b Fy(+)p 888 756 218 4 v 22 w(\(0)p Fs(;)17 b Fy(0\))26 b(=)p 1236 756 V 28 w(\(0)p Fs(;)17 b Fy(0\))k(+)h Fs(z)33 b Fy(=)27 b Fs(z)229 1011 y Fy(Ainsi,)f(on)f(v)m(oit)g(ais)m (\023)-46 b(emen)m(t)27 b(que)e(\()p Fu(Z)p Fs(;)17 b Fy(+\))25 b(est)h(un)f(group)s(e)f(et)h(que)h(l'on)f(p)s(eut)g(iden)m (ti\014er)83 1131 y Fu(N)37 b Fy(\022)-49 b(a)37 b(l'ensem)m(ble)j(des) c(\023)-46 b(el)m(\023)g(emen)m(ts)39 b(de)f Fu(Z)f Fy(de)h(la)f(forme) p 2094 1046 227 4 v 38 w(\(0)p Fs(;)17 b(n)p Fy(\))o(,)37 b(tout)g(en)h(pr)m(\023)-46 b(eserv)-5 b(an)m(t)39 b(les)83 1251 y(propri)m(\023)-46 b(et)m(\023)g(es)30 b(de)g(l'addition.)g (Ainsi)g(on)c(\023)-46 b(ecrira)29 b(par)g(exemple)j(simple)e(5)e Fp(2)g Fu(Z)p Fy(,)h(au)g(lieu)h(de)p 83 1287 218 4 v 83 1372 a(\(5)p Fs(;)17 b Fy(0\))27 b Fp(2)h Fu(Z)p Fy(.)83 1497 y Fx(D)n(\023)-54 b(e\014nition:)38 b Fy(P)m(our)33 b Fs(z)g Fp(2)28 b Fu(Z)p Fy(,)33 b(on)f(note)h Fp(\000)p Fs(z)38 b Fy(l'opp)s(os)m(\023)-46 b(e)33 b(de)g Fs(z)83 1692 y @beginspecial @setspecial @endspecial 169 x Fq(2.3)161 b(La)54 b(structure)d(d'anneau)83 1965 y @beginspecial @setspecial @endspecial 154 x Fw(2.3.1)136 b(v)l(o)t(cabulaire)83 2314 y Fx(D)n(\023)-54 b(e\014nition:)35 b Fy(Soit)29 b Fs(A)g Fy(un)h(ensem)m(ble,)i(+)d(et)g Fp(\002)h Fy(deux)h(loi)e(de)h (comp)s(osition)g(in)m(terne)g(sur)83 2434 y Fs(A)p Fy(.)j(On)f(dit)h (que)h(le)f(triplet)g(\()p Fs(A;)17 b Fy(+)p Fs(;)g Fp(\002)p Fy(\))32 b(est)h(un)g(anneau)g(si)229 2560 y({)49 b(\()p Fs(A;)17 b Fy(+\))40 b(est)i(un)g(group)s(e)f(comm)m(utatif)16 b(;)42 b(l')m(\023)-46 b(el)m(\023)g(emen)m(t)43 b(neutre)f(p)s(our)e (la)h(loi)g(+)g(est)327 2680 y(not)m(\023)-46 b(e)33 b(0.)229 2800 y({)49 b(\()p Fs(A;)17 b Fp(\002)p Fy(\))33 b(est)g(un)g(mono)-11 b(\177)-38 b(\020de)229 2921 y({)49 b(La)f(loi)f(de)i(comp)s(osition)g(in)m(terne)g Fp(\002)f Fy(est)h(distributiv)m(e)h(par)e(rapp)s(ort)f(\022)-49 b(a)48 b(+)g(:)327 3041 y Fp(8)p Fy(\()p Fs(p;)17 b(q)t(;)g(r)s Fy(\))27 b Fp(2)h Fs(A)883 3005 y Fm(3)1020 3041 y Fy(\()p Fs(p)17 b Fy(+)h Fs(q)t Fy(\))f Fp(\002)h Fs(r)31 b Fy(=)c Fs(p)18 b Fp(\002)g Fs(r)i Fy(+)e Fs(q)j Fp(\002)d Fs(r)s Fy(.)30 b(et)h Fp(8)p Fy(\()p Fs(p;)17 b(q)t(;)g(r)s Fy(\))27 b Fp(2)h Fs(A)2844 3005 y Fm(3)2981 3041 y Fs(r)20 b Fp(\002)e Fy(\()p Fs(p)g Fy(+)327 3162 y Fs(q)t Fy(\))k Fp(\002)g Fs(r)31 b Fy(=)c Fs(r)e Fp(\002)e Fs(p)f Fy(+)g Fs(r)j Fp(\002)d Fs(q)t Fy(.)83 3287 y Fx(D)n(\023)-54 b(e\014nition:)48 b Fy(On)42 b(dit)f(qu'un)i(anneau)f(comm)m(utativ)m (e)h(si)f(la)g(m)m(ultiplication)g(de)g(cet)83 3407 y(anneau)33 b(est)g(comm)m(utativ)m(e.)83 3573 y @beginspecial @setspecial @endspecial 154 x Fw(2.3.2)136 b(Propri)m(\023)-64 b(et)m(\023)g(es)46 b(de)f Fj(Z)229 3922 y Fy(On)31 b(admettra)f(qu'il)h(est)g(p)s(ossible) h(de)f(d)m(\023)-46 b(e\014nir)31 b(une)g(m)m(ultiplication)h Fp(\002)e Fy(sur)h Fu(Z)g Fy(telle)83 4042 y(que)474 4178 y Fp(8)p Fy(\()p Fs(a;)17 b(b)p Fy(\))23 b Fp(\002)f Fy(\()p Fs(c;)17 b(d)p Fy(\))27 b Fp(2)h Fy(\()p Fu(N)1307 4136 y Fm(2)1347 4178 y Fy(\))1385 4136 y Fm(2)p 1522 4093 547 4 v 1522 4178 a Fy(\()p Fs(a;)17 b(b)p Fy(\))22 b Fp(\002)h Fy(\()p Fs(c;)17 b(d)p Fy(\))27 b(=)p 2199 4093 731 4 v 27 w(\()p Fs(ac)22 b Fy(+)g Fs(bd;)17 b(bc)23 b Fy(+)f Fs(ad)p Fy(\))83 4369 y(et)33 b(telle)g(que)h(\()p Fu(Z)p Fs(;)17 b Fy(+)p Fs(;)g Fp(\002)p Fy(\))32 b(soit)h(un)g(anneau) g(comm)m(utatif.)229 4495 y(\(La)28 b(preuv)m(e)h(n'est)g(pas)f (particuli)m(\022)-46 b(eremen)m(t)30 b(di\016cile,)g(tous)e(les)h (ingr)m(\023)-46 b(edien)m(ts)29 b(on)m(t)d(\023)-46 b(et)m(\023)g(e)83 4615 y(donn)m(\023)g(es)42 b(ici,)f(mais)g(ce)g (n'est)h(pas)f(franc)m(hemen)m(t)h(passionan)m(t,)g(donc)f(on)g(p)s (ourra)f(s'en)83 4736 y(passer.\))83 4926 y @beginspecial @setspecial @endspecial 129 x Fw(2.3.3)136 b(R)m(\022)-64 b(egles)47 b(de)d(calcul)83 5250 y Fx(D)n(\023)-54 b(e\014nition:)45 b Fy(Soit)38 b(\()p Fs(A;)17 b Fy(+)p Fs(;)g Fp(\002)p Fy(\))37 b(un)i(anneau.)f(On)g(note)h(1)2232 5265 y Fn(A)2326 5250 y Fy(\(ou)f(1\))g(l')m(\023)-46 b(el)m(\023)g(emen)m(t)40 b(neutre)83 5370 y(p)s(our)c(la)g(m)m(ultiplication)i(dans)e Fs(A)p Fy(.)h(On)f(d)m(\023)-46 b(e\014nit)37 b(les)g(puissances)i(de)e Fs(a)f Fy(par)g Fs(a)2971 5334 y Fm(0)3045 5370 y Fy(=)e(1)i(et)p eop end %%Page: 18 22 TeXDict begin 18 21 bop 432 291 a @beginspecial @setspecial @endspecial Fy(18)2373 b Ft(CHAPITRE)34 b(2.)65 b Fu(Z)432 555 y Fy(la)32 b(r)m(\023)-46 b(ecurrence)35 b Fs(a)1063 519 y Fn(n)p Fm(+1)1228 555 y Fy(=)27 b Fs(a)1382 519 y Fn(n)1451 555 y Fp(\002)c Fs(a)p Fy(.)33 b(On)g(mon)m(tre)g(sans)g (di\016cult)m(\023)-46 b(e)1478 739 y Fp(8)p Fy(\()p Fs(n;)17 b(p)p Fy(\))27 b Fp(2)i Fu(N)1954 698 y Fm(2)1993 739 y Fs(a)2044 698 y Fn(n)p Fm(+)p Fn(p)2209 739 y Fy(=)f Fs(a)2364 698 y Fn(n)2433 739 y Fp(\002)23 b Fs(a)2584 698 y Fn(p)432 922 y Fy(Si)40 b Fs(a)f Fy(est)i(in)m(v)m(ersible,)i(on) c(d)m(\023)-46 b(e\014nit,)41 b(p)s(our)e Fs(n)h(>)g Fy(0)f Fs(a)2339 886 y Fo(\000)p Fn(n)2481 922 y Fy(par)h(la)f(r)m (\023)-46 b(ecurrence)42 b Fs(a)3307 886 y Fo(\000)p Fm(\()p Fn(n)p Fm(+1\))3594 922 y Fy(=)432 1043 y Fs(a)483 1007 y Fo(\000)p Fn(n)607 1043 y Fp(\002)23 b Fs(a)758 1007 y Fo(\000)p Fm(1)852 1043 y Fy(.)33 b(On)f(mon)m(tre)i(alors)e(de) h(m)m(^)-46 b(eme)1467 1226 y Fp(8)p Fy(\()p Fs(n;)17 b(p)p Fy(\))28 b Fp(2)g Fu(Z)1937 1185 y Fm(2)1977 1226 y Fs(a)2028 1185 y Fn(n)p Fm(+)p Fn(p)2193 1226 y Fy(=)f Fs(a)2347 1185 y Fn(n)2417 1226 y Fp(\002)22 b Fs(a)2567 1185 y Fn(p)2607 1226 y Fs(:)578 1410 y Fy(Il)36 b(n'est)h(pas)g (di\016cile)g(de)g(mon)m(trer)g(que)g(si)f Fs(a)h Fy(et)f Fs(b)g Fy(comm)m(uten)m(t)i(\()e(c'est)i(\022)-49 b(a)35 b(dire)i(si)432 1530 y Fs(ab)28 b Fy(=)g Fs(ba)p Fy(\),)33 b(alors)f Fp(8)p Fs(n)d Fp(2)f Fu(N)97 b Fy(\()p Fs(ab)p Fy(\))1652 1494 y Fn(n)1727 1530 y Fy(=)28 b Fs(a)1882 1494 y Fn(n)1929 1530 y Fs(b)1970 1494 y Fn(n)2018 1530 y Fy(.)432 1555 y @beginspecial @setspecial @endspecial 148 x Fx(Th)n(\023)-54 b(eor)n(\022)g(eme)34 b(8)f(\(Bin^)-56 b(ome)33 b(de)g(Newton\).)38 b Fz(Soit)31 b Fy(\()p Fs(A;)17 b Fy(+)p Fs(;)g Fp(\002)p Fy(\))31 b Fz(un)h(anne)-5 b(au.)31 b(Pour)g(tous)430 1823 y(\023)-47 b(el)n(\023)g(ements)32 b Fs(a)j Fz(et)g Fs(b)g Fz(de)g Fs(A)g Fz(tels)g(que)f Fs(ab)29 b Fy(=)e Fs(ba)p Fz(,)36 b(on)e(a)1498 2063 y Fy(\()p Fs(a)23 b Fy(+)f Fs(b)p Fy(\))1787 2022 y Fn(n)1862 2063 y Fy(=)2008 1956 y Fn(n)1970 1981 y Fk(X)1965 2164 y Fn(k)r Fm(=0)2110 1917 y Fk( )2176 1996 y Fs(n)2178 2132 y(k)2234 1917 y Fk(!)2300 2063 y Fs(a)2351 2022 y Fn(k)2394 2063 y Fs(b)2435 2022 y Fn(n)p Fo(\000)p Fn(k)2576 2063 y Fs(:)432 2313 y Fz(D)n(\023)-47 b(emonstr)-5 b(ation.)46 b Fy(On)41 b(v)-5 b(a)40 b(mon)m(trer)h(cela)h(par)e(r)m (\023)-46 b(ecurrence.)43 b(P)m(our)e Fs(n)g Fy(=)g(0,)g(l'iden)m(tit)m (\023)-46 b(e)432 2434 y(se)40 b(r)m(\023)-46 b(esure)41 b(\022)-49 b(a)40 b(1)f(=)h(1)g(tandis)g(que)h(p)s(our)f Fs(n)g Fy(=)g(1,)f(c'est)i(simplemen)m(t)h Fs(a)28 b Fy(+)f Fs(b)40 b Fy(=)g Fs(a)27 b Fy(+)g Fs(b)p Fy(.)432 2554 y(Supp)s(osons)43 b(donc)f(l'iden)m(tit)m(\023)-46 b(e)43 b(v)m(\023)-46 b(eri\014)m(\023)g(ee)44 b(au)d(rang)h Fs(n)p Fy(,)g(et)g(mon)m(trons)h(qu'elle)g(est)g(alors)432 2674 y(v)m(\023)-46 b(eri\014)m(\023)g(ee)34 b(au)e(rang)h Fs(n)22 b Fy(+)g(1.)542 2858 y(\()p Fs(a)g Fy(+)g Fs(b)p Fy(\))830 2817 y Fn(n)p Fm(+1)1050 2858 y Fy(=)83 b(\()p Fs(a)23 b Fy(+)f Fs(b)p Fy(\)\()p Fs(a)g Fy(+)g Fs(b)p Fy(\))1786 2817 y Fn(n)1050 3065 y Fy(=)83 b(\()p Fs(a)23 b Fy(+)f Fs(b)p Fy(\))1558 2958 y Fn(n)1519 2983 y Fk(X)1515 3166 y Fn(k)r Fm(=0)1660 2919 y Fk( )1726 2998 y Fs(n)1728 3134 y(k)1784 2919 y Fk(!)1850 3065 y Fs(a)1901 3024 y Fn(k)1943 3065 y Fs(b)1984 3024 y Fn(n)p Fo(\000)p Fn(k)1050 3357 y Fy(=)1252 3250 y Fn(n)1214 3275 y Fk(X)1209 3458 y Fn(k)r Fm(=0)1355 3211 y Fk( )1420 3290 y Fs(n)1422 3425 y(k)1479 3211 y Fk(!)1544 3357 y Fs(a)1595 3316 y Fn(k)r Fm(+1)1728 3357 y Fs(b)1769 3316 y Fn(n)p Fo(\000)p Fn(k)1932 3357 y Fy(+)2073 3250 y Fn(n)2035 3275 y Fk(X)2030 3458 y Fn(k)r Fm(=0)2176 3211 y Fk( )2241 3290 y Fs(n)2243 3425 y(k)2300 3211 y Fk(!)2365 3357 y Fs(a)2416 3316 y Fn(k)2459 3357 y Fs(b)2500 3316 y Fn(n)p Fo(\000)p Fn(k)r Fm(+1)1050 3649 y Fy(=)1252 3542 y Fn(n)1214 3566 y Fk(X)1209 3749 y Fn(k)r Fm(=0)1355 3503 y Fk( )1420 3581 y Fs(n)1422 3717 y(k)1479 3503 y Fk(!)1544 3649 y Fs(a)1595 3608 y Fn(k)r Fm(+1)1728 3649 y Fs(b)1769 3608 y Fm(\()p Fn(n)p Fm(+1\))p Fo(\000)p Fm(\()p Fn(k)r Fm(+1\))2222 3649 y Fy(+)2363 3542 y Fn(n)2325 3566 y Fk(X)2320 3749 y Fn(k)r Fm(=0)2466 3503 y Fk( )2531 3581 y Fs(n)2534 3717 y(k)2590 3503 y Fk(!)2655 3649 y Fs(a)2706 3608 y Fn(k)2749 3649 y Fs(b)2790 3608 y Fn(n)p Fo(\000)p Fn(k)r Fm(+1)1050 3949 y Fy(=)1209 3842 y Fn(n)p Fm(+1)1216 3867 y Fk(X)1211 4050 y Fn(k)r Fm(=1)1359 3803 y Fk( )1508 3882 y Fs(n)1425 4017 y(k)j Fp(\000)d Fy(1)1649 3803 y Fk(!)1715 3949 y Fs(a)1766 3908 y Fn(k)1809 3949 y Fs(b)1850 3908 y Fm(\()p Fn(n)p Fm(+1\))p Fo(\000)p Fn(k)2158 3949 y Fy(+)2299 3842 y Fn(n)2260 3867 y Fk(X)2256 4050 y Fn(k)r Fm(=0)2401 3803 y Fk( )2467 3882 y Fs(n)2469 4017 y(k)2525 3803 y Fk(!)2591 3949 y Fs(a)2642 3908 y Fn(k)2685 3949 y Fs(b)2726 3908 y Fn(n)p Fo(\000)p Fn(k)r Fm(+1)1050 4241 y Fy(=)83 b Fs(a)1260 4199 y Fn(n)p Fm(+1)1420 4241 y Fy(+)1561 4133 y Fn(n)1522 4158 y Fk(X)1518 4341 y Fn(k)r Fm(=1)1663 4094 y Fk( )1812 4173 y Fs(n)1729 4309 y(k)25 b Fp(\000)e Fy(1)1953 4094 y Fk(!)2019 4241 y Fs(a)2070 4199 y Fn(k)2113 4241 y Fs(b)2154 4199 y Fm(\()p Fn(n)p Fm(+1\))p Fo(\000)p Fn(k)2462 4241 y Fy(+)2603 4133 y Fn(n)2564 4158 y Fk(X)2560 4341 y Fn(k)r Fm(=1)2705 4094 y Fk( )2771 4173 y Fs(n)2773 4309 y(k)2829 4094 y Fk(!)2895 4241 y Fs(a)2946 4199 y Fn(k)2989 4241 y Fs(b)3030 4199 y Fn(n)p Fo(\000)p Fn(k)r Fm(+1)3283 4241 y Fy(+)f Fs(b)3422 4199 y Fn(n)p Fm(+1)1050 4532 y Fy(=)83 b Fs(a)1260 4491 y Fn(n)p Fm(+1)1420 4532 y Fy(+)1561 4425 y Fn(n)1522 4450 y Fk(X)1518 4633 y Fn(k)r Fm(=1)1663 4386 y Fk( )1878 4465 y Fs(n)1795 4601 y(k)25 b Fp(\000)d Fy(1)2019 4386 y Fk(!)2107 4532 y Fy(+)2205 4386 y Fk( )2271 4465 y Fs(n)2273 4601 y(k)2329 4386 y Fk(!!)2477 4532 y Fs(a)2528 4491 y Fn(k)2571 4532 y Fs(b)2612 4491 y Fm(\()p Fn(n)p Fm(+1\))p Fo(\000)p Fn(k)2920 4532 y Fy(+)g Fs(b)3059 4491 y Fn(n)p Fm(+1)1050 4824 y Fy(=)83 b Fs(a)1260 4783 y Fn(n)p Fm(+1)1398 4824 y Fs(b)22 b Fy(+)1602 4717 y Fn(n)1564 4742 y Fk(X)1559 4924 y Fn(k)r Fm(=1)1705 4678 y Fk( )1770 4756 y Fs(n)h Fy(+)f(1)1857 4892 y Fs(k)1997 4678 y Fk(!)2063 4824 y Fs(a)2114 4783 y Fn(k)2157 4824 y Fs(b)2198 4783 y Fm(\()p Fn(n)p Fm(+1\))p Fo(\000)p Fn(k)2506 4824 y Fy(+)g Fs(b)2645 4783 y Fn(n)p Fm(+1)1050 5124 y Fy(=)1209 5017 y Fn(n)p Fm(+1)1216 5042 y Fk(X)1211 5225 y Fn(k)r Fm(=0)1359 4978 y Fk( )1425 5057 y Fs(n)g Fy(+)g(1)1511 5192 y Fs(k)1652 4978 y Fk(!)1718 5124 y Fs(a)1769 5083 y Fn(k)1811 5124 y Fs(b)1852 5083 y Fm(\()p Fn(n)p Fm(+1\))p Fo(\000)p Fn(k)p 3599 5370 4 66 v 3603 5308 59 4 v 3603 5370 V 3661 5370 4 66 v eop end %%Page: 19 23 TeXDict begin 19 22 bop 83 291 a @beginspecial @setspecial @endspecial Ft(2.4.)65 b(EXER)m(CICES)2354 b Fy(19)83 456 y @beginspecial @setspecial @endspecial 99 x Fq(2.4)161 b(Exercices)83 637 y @beginspecial @setspecial @endspecial 202 774 a Fy(1.)49 b(Dans)25 b(l'ensem)m(ble)k(des)24 b(\023)-46 b(el)m(\022)g(ev)m(es)28 b(d'une)e(classe,)i(la)d(relation)h (binaire)g(\\)m(^)-46 b(etre)26 b(de)g(sexe)327 895 y(opp)s(os)m(\023) -46 b(e")33 b(est-elle)g(une)h(relation)e(d')m(\023)-46 b(equiv)-5 b(alence)16 b(?)83 930 y @beginspecial @setspecial @endspecial 202 1048 a(2.)49 b(On)32 b(note)1179 1168 y Fu(Z)p Fy([)p Fs(i)p Fy(])c(=)g Fp(f)p Fs(a)22 b Fy(+)g Fs(ib)p Fy(;)17 b(\()p Fs(a;)g(b)p Fy(\))28 b Fp(2)h Fu(Z)22 b Fp(\002)h Fu(Z)p Fp(g)p Fs(:)83 1229 y @beginspecial @setspecial @endspecial 368 1341 a Fy(\(a\))49 b(Mon)m(trer)36 b(que)h Fu(Z)p Fy([)p Fs(i)p Fy(])g(est)f(un)g(anneau)h(comm)m(utatif)f (\(on)g(admettra)g(que)h Fu(C)542 1461 y Fy(est)c(un)g(anneau)g(comm)m (utatif)7 b(\).)83 1502 y @beginspecial @setspecial @endspecial 363 1614 a(\(b\))49 b(P)m(our)43 b Fs(z)49 b Fy(=)c Fs(a)29 b Fy(+)g Fs(ib)46 b Fp(2)f Fu(Z)p Fy([)p Fs(i)p Fy(],)f(on)e(p)s(ose)h Fp(N)15 b Fy(\()p Fs(z)t Fy(\))45 b(=)g Fs(a)2458 1578 y Fm(2)2527 1614 y Fy(+)29 b Fs(b)2673 1578 y Fm(2)2713 1614 y Fy(.)42 b(Mon)m(trer)i(que)542 1735 y Fp(N)15 b Fy(\()p Fs(z)t(z)775 1698 y Fo(0)799 1735 y Fy(\))27 b(=)h Fp(N)15 b Fy(\()p Fs(z)t Fy(\))p Fp(N)g Fy(\()p Fs(z)1374 1698 y Fo(0)1397 1735 y Fy(\).)83 1776 y @beginspecial @setspecial @endspecial 374 1888 a(\(c\))49 b(Mon)m(trer)33 b(dans)g Fu(Z)g Fy(l'iden)m(tit)m(\023)-46 b(e)34 b(de)f(Lagrange)f(:) 1003 2103 y(\()p Fs(a)1092 2062 y Fm(2)1153 2103 y Fy(+)22 b Fs(b)1292 2062 y Fm(2)1332 2103 y Fy(\)\()p Fs(c)1450 2062 y Fm(2)1512 2103 y Fy(+)g Fs(d)1661 2062 y Fm(2)1700 2103 y Fy(\))28 b(=)f(\()p Fs(ac)22 b Fp(\000)h Fs(bd)p Fy(\))2252 2062 y Fm(2)2314 2103 y Fy(+)f(\()p Fs(ad)g Fy(+)g Fs(bc)p Fy(\))2793 2062 y Fm(2)2833 2103 y Fs(:)83 2223 y @beginspecial @setspecial @endspecial 363 2335 a Fy(\(d\))49 b(L'iden)m(tit)m(\023)-46 b(e)35 b(de)g(Lagrange)f (est-elle)i(vraie)f(dans)f(tout)h(anneau)f(comm)m(uta-)542 2455 y(tif)16 b(?)83 2472 y @beginspecial @setspecial @endspecial 374 2609 a(\(e\))49 b(Soit)25 b Fs(A)g Fy(un)h(anneau)g (comm)m(utatif.)g(On)f(note)h Fs(M)36 b Fy(l'ensem)m(ble)28 b(des)c(\023)-46 b(el)m(\023)g(emen)m(ts)542 2729 y(de)33 b Fs(A)f Fy(qui)i(p)s(euv)m(en)m(t)g(s')m(\023)-46 b(ecrire)34 b(comme)g(somme)g(de)f(deux)h(carr)m(\023)-46 b(es.)33 b(Mon)m(trer)542 2849 y(que)g(\()p Fs(M)5 b(;)17 b Fp(\002)p Fy(\))33 b(est)g(un)g(mono)-11 b(\177)-38 b(\020de.)83 2890 y @beginspecial @setspecial @endspecial 202 3002 a(3.)49 b(Soit)32 b Fs(j)i Fy(=)28 b(exp)q(\()902 2963 y Fm(2)p Fn(i\031)p 902 2979 103 4 v 935 3036 a Fm(3)1014 3002 y Fy(\).)k(Mon)m(trer)i(1)22 b(+)g Fs(j)28 b Fy(+)22 b Fs(j)1872 2966 y Fm(2)1939 3002 y Fy(=)27 b(0.)33 b(On)f(note)1166 3217 y Fu(Z)p Fy([)p Fs(j)6 b Fy(])28 b(=)g Fp(f)p Fs(a)22 b Fy(+)g Fs(j)6 b(b)p Fy(;)17 b(\()p Fs(a;)g(b)p Fy(\))28 b Fp(2)g Fu(Z)23 b Fp(\002)f Fu(Z)p Fp(g)p Fs(:)327 3432 y Fy(Mon)m(trer)28 b(que)h Fu(Z)p Fy([)p Fs(j)6 b Fy(])28 b(est)h(un)f(anneau)g(comm)m(utatif)g(\(on)g(admettra)g(que)g Fu(C)g Fy(est)g(un)327 3553 y(anneau)33 b(comm)m(utatif)7 b(\).)83 3594 y @beginspecial @setspecial @endspecial 202 3706 a(4.)49 b(On)35 b(dit)g(qu'en)e(\023)-46 b(el)m(\023)g(emen)m (t)36 b Fs(x)f Fy(d'un)h(anneau)f Fs(A)g Fy(est)g(nilp)s(oten)m(t)h(si) f(il)g(existe)i Fs(n)31 b Fp(\025)h Fy(0)327 3826 y(tel)c(que)g Fs(x)694 3790 y Fn(n)769 3826 y Fy(=)g(0.)f(Dans)g(ce)h(cas,)g(on)f (app)s(elle)h(indice)h(de)f(nilp)s(otence)h(de)e Fs(x)h Fy(le)g(plus)327 3947 y(p)s(etit)j Fs(n)g Fy(tel)g(que)h Fs(x)1024 3911 y Fn(n)1099 3947 y Fy(=)27 b(0.)k(Mon)m(trer)g(que)h (l'ensem)m(ble)i(des)29 b(\023)-46 b(el)m(\023)g(emen)m(ts)33 b(nilp)s(oten)m(ts)327 4067 y(d'un)g(anneau)g(comm)m(utatif)g(forme)g (un)g(anneau.)83 4084 y @beginspecial @setspecial @endspecial 202 4220 a(5.)49 b(Soit)32 b Fs(A)h Fy(un)g(anneau,)g Fs(e)g Fy(son)d(\023)-46 b(el)m(\023)g(emen)m(t)34 b(unit)m(\023)-46 b(e)33 b(p)s(our)g(la)f(m)m(ultiplication.)83 4256 y @beginspecial @setspecial @endspecial 368 4373 a(\(a\))49 b(Mon)m(trer)34 b(que)h(si)g(deux)d(\023)-46 b(el)m(\023)g(emen)m(ts)36 b Fs(a)e Fy(et)h Fs(b)f Fy(de)g Fs(A)h Fy(v)m(\023)-46 b(eri\014en)m(t)35 b Fs(ab)c Fy(=)f Fs(ba)p Fy(,)35 b(alors)542 4493 y(p)s(our)d Fs(n)c Fp(\025)g Fy(1,)k(on)h(a)1287 4774 y Fs(a)1338 4733 y Fn(n)1407 4774 y Fp(\000)23 b Fs(b)1548 4733 y Fn(n)1623 4774 y Fy(=)k(\()p Fs(a)c Fp(\000)f Fs(b)p Fy(\))2033 4667 y Fn(n)p Fo(\000)p Fm(1)2040 4692 y Fk(X)2035 4875 y Fn(k)r Fm(=0)2183 4774 y Fs(a)2234 4733 y Fn(k)2277 4774 y Fs(b)2318 4733 y Fn(n)p Fo(\000)p Fm(1)p Fo(\000)p Fn(k)2549 4774 y Fs(:)83 4978 y @beginspecial @setspecial @endspecial 363 5077 a Fy(\(b\))49 b(Mon)m(trer)33 b(que)g(si)g Fs(x)g Fy(est)h(nilp)s(oten)m(t,)f(alors)g Fs(e)22 b Fp(\000)h Fs(x)33 b Fy(est)g(in)m(v)m(ersible.)83 5119 y @beginspecial @setspecial @endspecial 374 5231 a(\(c\))49 b(Mon)m(trer)28 b(que)g(si)g Fs(x)g Fy(est)g(nilp)s(oten)m (t)h(et)e(que)i Fs(x)2168 5195 y Fo(0)2219 5231 y Fy(est)f(l'in)m(v)m (erse)i(de)e Fs(e)12 b Fp(\000)g Fs(x)p Fy(,)29 b(alors)542 5351 y Fs(e)22 b Fp(\000)g Fs(x)33 b Fy(est)h(nilp)s(oten)m(t.)p eop end %%Page: 20 24 TeXDict begin 20 23 bop 432 291 a @beginspecial @setspecial @endspecial Fy(20)2373 b Ft(CHAPITRE)34 b(2.)65 b Fu(Z)432 456 y @beginspecial @setspecial @endspecial 551 555 a Fy(6.)49 b(Soit)30 b Fs(A)g Fy(un)g(anneau,)h Fs(e)f Fy(son)e(\023)-46 b(el)m(\023)g(emen)m(t)32 b(unit)m(\023)-46 b(e.)31 b(P)m(our)f Fs(x)p Fy(,)e(\023)-46 b(el)m(\023)g(emen)m(t)32 b(nilp)s(oten)m(t)e(d'in-)676 676 y(dice)j Fs(p)p Fy(,)g(on)f(p)s(ose) 1837 853 y(exp)q(\()p Fs(x)p Fy(\))c(=)2295 740 y Fn(p)2253 771 y Fk(X)2248 953 y Fn(k)r Fm(=0)2403 785 y Fs(x)2458 749 y Fn(p)p 2403 830 96 4 v 2413 921 a Fs(p)p Fy(!)676 1092 y(Soien)m(t)k Fs(a)f Fy(et)h Fs(b)f Fy(deux)f(\023)-46 b(el)m(\023)g(emen)m(ts)33 b(nilp)s(oten)m(ts)g(de)f Fs(a)f Fy(P)m(ourquoi)i(a-t'on)d(le)i(droit)f(de)676 1212 y(parler)e(de)h(exp)q(\()p Fs(a)16 b Fy(+)g Fs(b)p Fy(\))g(?)29 b(Mon)m(trer)h(que)h(exp)q(\()p Fs(a)16 b Fy(+)g Fs(b)p Fy(\))27 b(=)g(exp)q(\()p Fs(a)p Fy(\))17 b(exp)q(\()p Fs(b)p Fy(\).)30 b(Mon)m(trer)676 1333 y(que)j(p)s(our)f (tout)h(nilp)s(oten)m(t)g Fs(x)g Fy(exp)q(\()p Fs(x)p Fy(\))g(est)g(in)m(v)m(ersible.)432 1375 y @beginspecial @setspecial @endspecial 551 1487 a(7.)49 b Fz(Th)n(\023)-47 b(eor)n(\022)g(eme)32 b(de)j(L)-5 b(agr)g(ange)39 b Fy(Soit)34 b Fs(G)f Fy(un)g(group)s(e)g(\014ni.)h(et)f Fs(H)41 b Fy(un)34 b(sous-group)s(e)f(de)676 1607 y Fs(G)p Fy(.)f(On)h(d)m(\023) -46 b(e\014nit)33 b(une)g(relation)g(binaire)g Fp(R)2229 1622 y Fn(H)2330 1607 y Fy(sur)g Fs(G)f Fy(par)1471 1825 y(\()p Fs(g)t Fp(R)1644 1840 y Fn(H)1711 1825 y Fs(g)1762 1784 y Fo(0)1785 1825 y Fy(\))55 b Fp(\()-17 b(\))55 b(9)p Fs(h)28 b Fp(2)g Fs(H)105 b(g)2586 1784 y Fo(0)2637 1825 y Fy(=)27 b Fs(g)t(h:)432 1930 y @beginspecial @setspecial @endspecial 717 2042 a Fy(\(a\))48 b(Mon)m(trer)34 b(que)f Fp(R)1535 2057 y Fn(H)1635 2042 y Fy(est)h(une)f(relation)f(d')m(\023) -46 b(equiv)-5 b(alence.)432 2084 y @beginspecial @setspecial @endspecial 711 2196 a(\(b\))49 b(Mon)m(trer)33 b(qu'il)g(y)g(a)f (exactemen)m(t)i Fp(j)p Fs(H)8 b Fp(j)31 b Fy(classes)k(d')m(\023)-46 b(equiv)-5 b(alences)35 b(p)s(our)d Fp(R)3602 2211 y Fn(H)890 2317 y Fy(et)h(qu'elles)i(on)m(t)d(toutes)i(le)f(m)m(^)-46 b(eme)33 b(cardinal.)432 2353 y @beginspecial @setspecial @endspecial 722 2471 a(\(c\))49 b(En)33 b(d)m(\023)-46 b(eduire)34 b(que)g Fp(j)p Fs(H)8 b Fp(j)31 b Fy(divise)j Fp(j)p Fs(G)p Fp(j)p Fy(.)432 2512 y @beginspecial @setspecial @endspecial 711 2624 a(\(d\))49 b(Soit)36 b Fs(x)e Fp(2)f Fs(G)p Fy(.)j(Mon)m(trer)h(qu'il)f(existe)i Fs(n)e Fy(en)m(tier)h (naturel)f(non)g(n)m(ul)g(tel)h(que)890 2745 y Fs(x)945 2709 y Fn(n)1020 2745 y Fy(=)28 b Fs(e)p Fy(.)k(Indication)h(:)f (Consid)m(\023)-46 b(erer)34 b(la)d(suite)j Fs(e;)17 b(x;)g(x)2822 2709 y Fm(2)2861 2745 y Fs(;)g(x)2960 2709 y Fm(3)3000 2745 y Fs(;)g(:)g(:)g(:)48 b Fy(On)32 b(app)s(elle)890 2865 y(ordre)44 b(de)g Fs(x)g Fy(le)g(plus)g(p)s(etit)g(en)m(tier)g (naturel)g(non)g(n)m(ul)g(tel)g(que)g Fs(x)3381 2829 y Fn(n)3475 2865 y Fy(=)i Fs(e)p Fy(.)890 2986 y(Mon)m(trer)34 b(que)f(l'ordre)g(de)g Fs(x)g Fy(divise)h Fp(j)p Fs(G)p Fp(j)p Fy(.)432 3027 y @beginspecial @setspecial @endspecial 551 3139 a(8.)49 b Fz(Th)n(\023)-47 b(eor)n(\022)g(eme)31 b(de)k(Cauchy)676 3260 y Fy(\()p Fs(G;)17 b(?)p Fy(\))37 b(d)m(\023)-46 b(esigne)40 b(un)f(group)s(e)f(\014ni,)h(d'ordre)g Fs(n)p Fy(,)f(d')m(\023)-46 b(el)m(\023)g(emen)m(t)41 b(neutre)e Fs(e)p Fy(.)g Fs(p)f Fy(est)h(un)676 3380 y(diviseur)32 b(premier)f(de)g Fs(n)p Fy(.)f(On)g(d)m(\023)-46 b(esigne)32 b(par)e Fs(\033)k Fy(la)c(bijection)h(de)g Fp(f)p Fy(1)p Fs(;)17 b(:)g(:)g(:)e(;)i(p)p Fp(g)30 b Fy(dans)676 3500 y(lui-meme)j(d)m(\023)-46 b(e\014nie)34 b(de)f(la)g(mani)m(\022)-46 b(ere)33 b(suiv)-5 b(an)m(te)34 b(:)f Fs(\033)t Fy(\()p Fs(p)p Fy(\))27 b(=)h(1)k(et)1461 3718 y Fp(8)p Fs(i)c Fp(2)g(f)p Fy(1)p Fs(;)17 b(:)g(:)g(:)f(;)h(p)22 b Fp(\000)g Fy(1)p Fp(g)97 b Fs(\033)t Fy(\()p Fs(i)p Fy(\))28 b(=)g Fs(i)22 b Fy(+)g(1)p Fs(:)676 3935 y Fy(On)32 b(d)m(\023)-46 b(e\014nit,)34 b(par)f(r)m(\023)-46 b(ecurrence,)35 b Fs(\033)1901 3899 y Fn(k)1976 3935 y Fy(p)s(our)e(tout)g(en)m(tier)h (naturel)f Fs(k)j Fy(par)d(:)g Fs(\033)3409 3899 y Fm(0)3476 3935 y Fy(=)28 b(Id)676 4056 y(et)k Fs(\033)848 4020 y Fn(k)r Fm(+1)1009 4056 y Fy(=)27 b Fs(\033)g Fp(\016)22 b Fs(\033)1325 4020 y Fn(k)1367 4056 y Fy(.)432 4072 y @beginspecial @setspecial @endspecial 717 4209 a(\(a\))48 b(V)m(\023)-46 b(eri\014er)40 b(que,)g(si)g Fs(r)i Fy(d)m(\023)-46 b(esigne)40 b(le)g(reste)g(de)f(la)g(division)h(euclidienne)i(de)d Fs(k)890 4330 y Fy(par)33 b Fs(p)p Fy(,)f(on)h(a)f Fs(\033)1448 4294 y Fn(k)1518 4330 y Fy(=)c Fs(\033)1681 4294 y Fn(r)1719 4330 y Fy(.)432 4366 y @beginspecial @setspecial @endspecial 711 4484 a(\(b\))49 b(On)33 b(d)m(\023)-46 b(e\014nit)33 b(une)h(partie)e Fs(G)1899 4448 y Fn(p)1971 4484 y Fy(de)i Fs(E)k Fy(par)1367 4701 y Fs(E)c Fy(=)28 b Fp(f)p Fy(\()p Fs(x)1720 4716 y Fm(1)1759 4701 y Fs(;)17 b(:)g(:)g(:)f(;)h(x)2033 4716 y Fn(p)2073 4701 y Fy(\))27 b Fp(2)i Fs(G)2310 4660 y Fn(p)2447 4701 y Fs(x)2502 4716 y Fm(1)2564 4701 y Fs(?)22 b Fp(\001)17 b(\001)g(\001)j Fs(?)i(x)2899 4716 y Fn(p)2967 4701 y Fy(=)27 b Fs(e)p Fp(g)p Fs(:)432 4826 y @beginspecial @setspecial @endspecial 722 4935 a Fy(\(c\))49 b(On)33 b(d)m(\023)-46 b(e\014nit)33 b(une)h(relation)e(binaire)h Fp(R)g Fy(sur)g Fs(E)39 b Fy(par)890 5068 y Fk(\200)934 5153 y Fy(\()p Fs(x)1027 5168 y Fm(1)1067 5153 y Fs(;)17 b(:)g(:)g(:)f(x)1297 5168 y Fn(p)1337 5153 y Fy(\))p Fp(R)p Fy(\()p Fs(x)1552 5112 y Fo(0)1552 5178 y Fm(1)1592 5153 y Fs(;)h(:)g(:)g(:)f(x)1822 5112 y Fo(0)1822 5178 y Fn(p)1862 5153 y Fy(\))1900 5068 y Fk(\212)1999 5153 y Fp(\()-17 b(\))2237 5068 y Fk(\200)2281 5153 y Fp(9)p Fs(k)31 b Fp(2)d Fu(N)97 b Fp(8)p Fs(i)29 b Fp(2)f(f)p Fy(1)p Fs(;)17 b(:)g(:)g(:)e(;)i(p)p Fp(g)p Fs(x)3363 5112 y Fo(0)3363 5178 y Fn(i)3419 5153 y Fy(=)28 b Fs(x)3578 5172 y Fn(\033)3620 5153 y Fh(k)3659 5172 y Fm(\()p Fn(i)p Fm(\))3742 5068 y Fk(\212)3786 5153 y Fs(:)432 5271 y @beginspecial @setspecial @endspecial 970 5370 a Fy(i.)49 b(V)m(\023)-46 b(eri\014er)33 b(que)h Fp(R)f Fy(est)g(une)g(relation)g (d')m(\023)-46 b(equiv)-5 b(alence)35 b(sur)e Fs(E)6 b Fy(.)p eop end %%Page: 21 25 TeXDict begin 21 24 bop 83 291 a @beginspecial @setspecial @endspecial Ft(2.4.)65 b(EXER)m(CICES)2354 b Fy(21)83 456 y @beginspecial @setspecial @endspecial 594 555 a(ii.)49 b(V)m(\023)-46 b(eri\014er)37 b(que)f(les)h(classes)g(d')m(\023)-46 b(equiv)-5 b(alences)39 b(p)s(our)c(la)h(relation)g Fp(R)g Fy(on)m(t)724 676 y(c)m(hacune)e(1)e(ou)h Fs(p)d Fy(\023)-46 b(el)m(\023)g(emen)m(ts.)83 712 y @beginspecial @setspecial @endspecial 363 830 a(\(d\))49 b(On)c(note)g Fs(\013)h Fy(le)g(nom)m(bre)h(de)f(classes)h(p)s(our)e Fp(R)h Fy(a)m(y)m(an)m(t)g (1)c(\023)-46 b(el)m(\023)g(emen)m(t,)48 b Fs(\014)i Fy(le)542 951 y(nom)m(bre)33 b(de)g(classes)i(p)s(our)d Fp(R)h Fy(a)m(y)m(an)m(t)g Fs(p)d Fy(\023)-46 b(el)m(\023)g(emen)m(ts.) 35 b(Mon)m(trer)e(que)1612 1171 y Fs(\013)23 b Fy(+)f Fs(\014)6 b(p)27 b Fy(=)g Fs(n)2093 1130 y Fn(p)p Fo(\000)p Fm(1)2223 1171 y Fs(:)83 1290 y @beginspecial @setspecial @endspecial 374 1408 a Fy(\(e\))49 b(En)41 b(d)m(\023)-46 b(eduire)42 b(qu'il)f(existe)i(un)38 b(\023)-46 b(el)m(\023)g(emen)m(t) 43 b Fs(x)e Fy(de)g Fs(G)g Fy(distinct)h(de)f Fs(e)g Fy(tel)g(que)542 1528 y Fs(x)597 1492 y Fn(?p)700 1528 y Fy(=)27 b Fs(e)p Fy(.)83 1545 y @beginspecial @setspecial @endspecial 202 1683 a(9.)49 b(On)32 b(consid)m(\022)-46 b(ere)35 b(l'ensem)m(ble)1187 1903 y Fs(G)28 b Fy(=)g Fp(f)p Fs(x)22 b Fy(+)g Fs(y)1673 1816 y Fp(p)p 1755 1816 49 4 v 1755 1903 a Fy(2;)17 b Fs(x)1903 1862 y Fm(2)1965 1903 y Fp(\000)22 b Fy(2)p Fs(y)2165 1862 y Fm(2)2231 1903 y Fy(=)28 b(1)p Fp(g)p Fs(:)83 2010 y @beginspecial @setspecial @endspecial 368 2123 a Fy(\(a\))49 b(Mon)m(trer)33 b(que)g Fs(G)g Fy(est)g(un)g(group)s(e.)83 2165 y @beginspecial @setspecial @endspecial 363 2278 a(\(b\))49 b(Soien)m(t)36 b Fs(z)41 b Fy(et)36 b Fs(z)1092 2241 y Fo(0)1152 2278 y Fy(dans)h Fs(G)p Fy(,)f(a)m(v)m(ec)h Fs(z)i Fy(=)33 b Fs(x)25 b Fy(+)f Fs(y)2166 2196 y Fp(p)p 2249 2196 V 82 x Fy(2)35 b(et)h Fs(z)2499 2241 y Fo(0)2557 2278 y Fy(=)d Fs(x)2721 2241 y Fo(0)2770 2278 y Fy(+)24 b Fs(y)2922 2241 y Fo(0)2944 2196 y Fp(p)p 3027 2196 V 82 x Fy(2,)36 b(a)m(v)m(ec)542 2398 y Fs(x)f(>)g(x)798 2362 y Fo(0)857 2398 y Fs(>)f Fy(1,)j Fs(y)h(>)d Fy(0)i(et)g Fs(y)1533 2362 y Fo(0)1590 2398 y Fs(>)e Fy(0.)i(On)g(p)s(ose)g Fs(t)e Fy(=)g Fs(z)t Fy(\()p Fs(z)2522 2362 y Fo(0)2547 2398 y Fy(\))2585 2362 y Fo(\000)p Fm(1)2679 2398 y Fy(.)i(Soien)m(t)h Fs(\013)f Fy(et)g Fs(\014)542 2518 y Fy(en)m(tiers)d(tels)f(que)h Fs(t)27 b Fy(=)h Fs(\013)23 b Fy(+)f Fs(\014)1625 2436 y Fp(p)p 1707 2436 V 1707 2518 a Fy(2.)33 b(Mon)m(trer)g(que)h(1)27 b Fs(<)h(\013)g(<)g(x)33 b Fy(et)f Fs(\014)i(>)27 b Fy(0.)83 2555 y @beginspecial @setspecial @endspecial 374 2673 a(\(c\))49 b(Quel)28 b(est)g(le)g(plus)h(p)s(etit)f(en)m(tier)h Fs(x)f Fy(tel)g(qu'il)h(existe)g Fs(y)i(>)c Fy(0)h(a)m(v)m(ec)h Fs(x)12 b Fy(+)g Fs(y)3095 2591 y Fp(p)p 3178 2591 V 82 x Fy(2)27 b Fp(2)542 2793 y Fs(G)16 b Fy(?)83 2811 y @beginspecial @setspecial @endspecial 363 2948 a(\(d\))49 b(Mon)m(trer)33 b(que)g Fs(G)28 b Fy(=)g Fp(f)p Fy(\(3)21 b(+)h(2)1616 2866 y Fp(p)p 1699 2866 V 82 x Fy(2\))1786 2912 y Fn(n)1833 2948 y Fy(;)17 b Fs(n)27 b Fp(2)h Fu(Z)p Fp(g)p Fy(.)83 2990 y @beginspecial @setspecial @endspecial 374 3103 a(\(e\))49 b(D)m(\023)-46 b(ecrire)22 b(l'ensem)m(ble)k(des)d (p)s(oin)m(ts)g(\022)-49 b(a)22 b(co)s(ordonn)m(\023)-46 b(ees)24 b(en)m(ti)m(\022)-46 b(eres)24 b(de)f(l'h)m(yp)s(erb)s(ole)542 3223 y(d')m(\023)-46 b(equation)1650 3344 y Fs(x)1705 3302 y Fm(2)1767 3344 y Fp(\000)22 b Fy(2)p Fs(y)1967 3302 y Fm(2)2033 3344 y Fy(=)28 b(1)p Fs(:)83 3417 y @beginspecial @setspecial @endspecial 153 3535 a Fy(10.)49 b(Soit)33 b Fs(u)g Fy(un)g(endomorphisme)j(de)d Fu(R)1661 3499 y Fm(2)1701 3535 y Fy(.)g(On)g(supp)s(ose)i(que)f Fs(u)f Fy(est)h(un)g(pro)5 b(jecteur,)327 3656 y(c'est)34 b(\022)-49 b(a)32 b(dire)h(que)h Fs(u)21 b Fp(\016)h Fs(u)27 b Fy(=)h Fs(u)p Fy(.)k(On)h(note)1155 3876 y Fs(V)49 b Fy(=)28 b Fp(f)p Fs(a)p Fy(Id)22 b(+)g Fs(bu)p Fy(;)17 b(\()p Fs(a;)g(b)p Fy(\))28 b Fp(2)g Fu(R)22 b Fp(\002)h Fu(R)p Fp(g)p Fs(:)327 4096 y Fy(Mon)m(trer)35 b(que)f Fs(V)22 b Fy(,)34 b(m)m(uni)h(de)f(l'addition)g(des)h (endomorphismes)i(et)d(de)g(la)g(com-)327 4216 y(p)s(osition)f(comme)g (loi)g(m)m(utliplicativ)m(e,)i(est)e(un)g(anneau.)p eop end %%Page: 22 26 TeXDict begin 22 25 bop 432 291 a @beginspecial @setspecial @endspecial Fy(22)2373 b Ft(CHAPITRE)34 b(2.)65 b Fu(Z)p eop end %%Page: 23 27 TeXDict begin 23 26 bop 83 291 a @beginspecial @setspecial @endspecial 165 x @beginspecial @setspecial @endspecial 767 x Fv(Chapitre)77 b(3)83 1642 y(Divisibilit)-5 b(\023)-111 b(e)83 1974 y @beginspecial @setspecial @endspecial 186 x Fq(3.1)161 b(Sous-group)t(es)50 b(de)j Fl(Z)229 2386 y Fy(P)m(our)29 b(tout)e(en)m(tier)i(naturel)g(n)m(ul)f Fs(n)p Fy(,)h(on)e(note)h Fs(n)p Fu(Z)h Fy(l'ensem)m(ble)i(des)d(m)m (ultiples)i(de)f Fs(n)f Fy(:)1312 2641 y Fs(n)p Fu(Z)g Fy(=)f Fp(f)p Fs(k)s(n)p Fy(;)17 b Fs(k)31 b Fp(2)d Fu(Z)p Fp(g)p Fs(:)83 2732 y @beginspecial @setspecial @endspecial 130 x Fx(Th)n(\023)-54 b(eor)n(\022)g(eme)37 b(9.)j Fz(L)-5 b(es)34 b(ensembles)e Fs(A)h Fz(tel)h(que)g Fy(\()p Fs(A;)17 b Fy(+\))33 b Fz(soit)g(un)h(sous-gr)-5 b(oup)g(e)33 b(de)g Fy(\()p Fu(Z)p Fs(;)17 b Fy(+\))83 2982 y Fz(sont)43 b(exactement)g(les)g(ensembles)e(de)i(la)g(forme)g Fs(n)p Fu(Z)p Fz(,)h(o)q(\022)-51 b(u)43 b Fs(n)h Fz(d)n(\023)-47 b(ecrit)42 b(l'ensemble)f(des)83 3103 y(entiers)35 b(natur)-5 b(els.)83 3324 y(D)n(\023)-47 b(emonstr)-5 b(ation.)193 b Fy({)48 b(Sens)28 b(direct)g(:)f(tout)g(sous-group)s(e)g(de)g Fu(Z)g Fy(est)h(de)f(la)g(forme)g Fs(n)p Fu(Z)p Fy(.)327 3444 y(Soit)39 b Fs(A)h Fy(un)g(sous-group)s(e)f(de)h Fu(Z)p Fy(.)g(Si)g Fs(A)f Fy(est)i(r)m(\023)-46 b(eduit)40 b(\022)-49 b(a)39 b Fp(f)p Fy(0)p Fp(g)p Fy(,)g(on)g(a)g(termin)m(\023) -46 b(e)41 b(car)327 3565 y Fp(f)p Fy(0)p Fp(g)g Fy(=)h(0)p Fu(Z)p Fy(.)f(Sinon,)g Fs(A)g Fy(con)m(tien)m(t)h(au)f(moins)h(un)c (\023)-46 b(el)m(\023)g(emen)m(t)43 b(non)e(n)m(ul)g(\(metton)m(t)327 3685 y Fs(x)p Fy(\).)h(Lorsque)h Fs(x)g Fy(est)f(dans)h Fs(A)p Fy(,)f Fp(\000)p Fs(x)h Fy(y)f(est)h(aussi.)g(Mais)g(de)f Fs(x)h Fy(et)f(de)g Fp(\000)p Fs(x)p Fy(,)h(l'un)327 3806 y(des)30 b(deux)g(est)f(stricemen)m(t)i(p)s(ositif.)e(On)g(en)g(d) m(\023)-46 b(eduit)30 b(que)f(l'ensem)m(ble)j Fs(A)14 b Fp(\\)g Fu(N)3132 3821 y Fo(\003)3201 3806 y Fy(est)327 3926 y(non-vide.)32 b(Soit)g(donc)g Fs(n)1241 3941 y Fm(0)1313 3926 y Fy(le)g(plus)g(p)s(etit)d(\023)-46 b(el)m(\023)g(emen) m(t)34 b(de)e Fs(A)20 b Fp(\\)h Fu(N)2596 3941 y Fo(\003)2635 3926 y Fy(.)32 b(On)g(v)-5 b(a)31 b(mon)m(trer)327 4046 y(que)i Fs(A)28 b Fy(=)g Fs(n)771 4061 y Fm(0)810 4046 y Fu(Z)p Fy(.)327 4167 y(On)h(mon)m(tre)h(par)f(r)m(\023)-46 b(ecurrence)31 b(sur)f Fs(k)i Fy(que)e(p)s(our)f(tout)g Fs(k)i Fp(2)d Fu(N)p Fy(,)h Fs(k)s(n)2726 4182 y Fm(0)2793 4167 y Fp(2)g Fs(A)g Fy(\(utiliser)327 4287 y(l'iden)m(tit)m(\023)-46 b(e)40 b(\()p Fs(k)28 b Fy(+)e(1\))p Fs(n)1106 4302 y Fm(0)1182 4287 y Fy(=)37 b Fs(k)s(n)1407 4302 y Fm(0)1473 4287 y Fy(+)26 b Fs(n)1633 4302 y Fm(0)1672 4287 y Fy(\).)38 b(On)g(en)h(d)m(\023)-46 b(eduit)39 b(que)g(p)s(our)f(tout)f Fs(k)j Fp(2)e Fu(N)p Fy(,)327 4407 y Fp(\000)p Fs(k)s(n)516 4422 y Fm(0)584 4407 y Fp(2)28 b Fs(A)p Fy(.)33 b(Finalemen)m(t)g Fs(n)1379 4422 y Fm(0)1419 4407 y Fu(Z)28 b Fp(\032)g Fs(A)p Fy(.)327 4528 y(Soit)c Fs(n)h Fy(un)d(\023)-46 b(el)m(\023)g(emen)m(t)27 b(de)e Fs(A)6 b Fp(\\)g Fu(N)1433 4543 y Fo(\003)1472 4528 y Fy(.)25 b(On)g(e\013ectue)h(la)e(division)i (euclidienne)h(de)e Fs(n)g Fy(par)327 4648 y Fs(n)385 4663 y Fm(0)458 4648 y Fy(:)33 b Fs(n)c Fy(=)g Fs(n)768 4663 y Fm(0)808 4648 y Fs(q)d Fy(+)d Fs(r)36 b Fy(a)m(v)m(ec)f(0)28 b Fp(\024)i Fs(r)i(<)c(n)1693 4663 y Fm(0)1733 4648 y Fy(.)33 b(On)h(a)f(n)m(\023)-46 b(ecessairemen)m(t)37 b Fs(q)32 b(>)d Fy(0,)k(sinon)h(le)327 4769 y(caract)m(\022)-46 b(ere)32 b(minimal)h(de)f Fs(n)1309 4784 y Fm(0)1381 4769 y Fy(serait)g(con)m(tredit.)h(Il)f(s'ensuit)h(que)g(0)28 b Fp(\024)g Fs(r)i(<)e(n)p Fy(.)k(On)327 4889 y(a)27 b Fs(r)j Fy(=)e Fs(n)11 b Fy(+)g(\()p Fp(\000)p Fs(n)910 4904 y Fm(0)950 4889 y Fs(q)t Fy(\).)27 b Fs(n)h Fy(et)f Fp(\000)p Fs(n)1418 4904 y Fm(0)1458 4889 y Fs(q)k Fy(son)m(t)d(dans)g Fs(A)p Fy(,)f(or)g Fs(A)g Fy(est)h(un)g(group)s(e)f(donc)g Fs(r)j Fy(est)327 5009 y(dans)g Fs(A)p Fy(.)f Fs(r)i Fy(est)f(p)s(ositif,)f(il)g(est)h(dans)g Fs(A)f Fy(et)g(est)h (strictemen)m(t)h(plus)f(p)s(etit)f(que)h Fs(n)3225 5024 y Fm(0)3294 5009 y Fy(:)327 5130 y(il)38 b(ne)g(p)s(eut)e(^)-46 b(etre)38 b(strictemen)m(t)i(p)s(ositif,)d(sinon)i(cela)f(con)m (tredirait)h(le)f(caract)m(\022)-46 b(ere)327 5250 y(minimal)44 b(de)h Fs(n)917 5265 y Fm(0)1000 5250 y Fy(:)f(on)f(a)h(donc)g Fs(r)49 b Fy(=)e(0,)c(soit)h Fs(n)j Fy(=)g Fs(n)2371 5265 y Fm(0)2411 5250 y Fs(q)t Fy(,)c(ce)i(qui)f(signi\014e)h(que)327 5370 y Fs(n)34 b Fp(2)h Fs(n)578 5385 y Fm(0)617 5370 y Fu(Z)p Fy(.)i(Reste)g(a)m(v)m(oir)g(les)e(\023)-46 b(el)m(\023)g(emen)m(ts)38 b(n)m(\023)-46 b(egatifs)37 b(:)f(soit)h Fs(n)f Fy(un)e(\023)-46 b(el)m(\023)g(emen)m(t)38 b(n)m(\023)-46 b(egatif)1653 5620 y(23)p eop end %%Page: 24 28 TeXDict begin 24 27 bop 432 291 a @beginspecial @setspecial @endspecial Fy(24)1795 b Ft(CHAPITRE)34 b(3.)65 b(DIVISIBILIT)3615 265 y(\023)3602 291 y(E)676 555 y Fy(de)33 b Fs(A)16 b Fy(;)34 b Fp(\000)p Fs(n)g Fy(est)g(un)d(\023)-46 b(el)m(\023)g(emen) m(t)35 b(p)s(ositif)e(de)h Fs(A)p Fy(,)f(donc)h Fp(\000)p Fs(n)c Fp(2)f Fs(n)2896 570 y Fm(0)2936 555 y Fu(Z)p Fy(,)k(donc)h Fs(n)29 b Fp(2)g Fs(n)3536 570 y Fm(0)3576 555 y Fu(Z)p Fy(.)676 676 y(Ainsi)k Fs(A)28 b Fp(\032)g Fs(n)1192 691 y Fm(0)1232 676 y Fu(Z)p Fy(,)33 b(ce)g(qui)g(ac)m(h)m (\022)-46 b(ev)m(e)35 b(la)d(preuv)m(e.)578 796 y({)49 b(R)m(\023)-46 b(ecipro)s(que)33 b(:)g(tout)f(ensem)m(ble)j(de)f(la)e (forme)h Fs(n)p Fu(Z)g Fy(est)g(un)g(sous-group)s(e)g(de)g Fu(Z)p Fy(.)676 916 y(La)f(preuv)m(e)i(est)f(laiss)m(\023)-46 b(ee)35 b(en)e(exercice.)p 3599 1038 4 66 v 3603 975 59 4 v 3603 1038 V 3661 1038 4 66 v 432 1194 a @beginspecial @setspecial @endspecial 184 x Fq(3.2)160 b(PGCD)432 1478 y @beginspecial @setspecial @endspecial 151 x Fw(3.2.1)136 b(Compl)m(\023)-64 b(emen)l(ts)46 b(sur)f(les)g(group)t(es)432 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y(group)s(e)51 b(engendr)m(\023)-46 b(e)52 b(par)f Fs(X)58 b Fy(l'in)m(tersection)53 b(de)f(tous)f(les)h(sous-group)s(es)g(de)f Fs(G)g Fy(qui)432 2712 y(con)m(tiennen)m(t)f Fs(X)55 b Fy(\(il)48 b(existe)h(au)f(moins)g (un)g(tel)h(sous-group)s(e,)f(gar)f Fs(G)g Fy(en)i(est)f(un\).)432 2833 y(D'apr)m(\022)-46 b(es)22 b(le)g(lemme)h(pr)m(\023)-46 b(ec)m(\023)g(eden)m(t,)24 b(c'est)f(bien)f(un)g(sous-group)s(e)g(de)h (\()p Fs(G;)17 b Fp(\002)p Fy(\).)22 b(P)m(ar)g(construc-)432 2953 y(tion,)k(c'est)i(le)e(plus)i(p)s(etit)e(sous-group)s(e)h(de)g Fs(G)f Fy(qui)h(con)m(tienne)h Fs(X)8 b Fy(.)26 b(On)g(le)h(note)g Fs(<)g(X)36 b(>)p Fy(.)432 3075 y Fx(D)n(\023)-54 b(e\014nition:)56 b Fy(Si)49 b Fs(x)f Fy(est)h(un)d(\023)-46 b(el)m(\023)g(emen)m(t)50 b(d'un)f(group)s(e)g Fs(G)p Fy(,)f(on)g(app)s(elle)h(sous-group)s(e)432 3195 y(engendr)m(\023)-46 b(e)35 b(par)g Fs(x)g Fy(et)f(on)h(note)f (simplemen)m(t)j Fs(<)31 b(x)g(>)j Fy(le)h(sous-group)s(e)g(de)g Fs(G)f Fy(engendr)m(\023)-46 b(e)432 3316 y(par)32 b(le)h(singleton)h Fp(f)p Fs(x)p Fp(g)p Fy(.)432 3437 y Fx(Exemple:)k Fy(P)m(our)33 b Fs(n)28 b Fp(2)g Fu(Z)33 b Fs(<)28 b(n)f(>)p Fy(=)h Fs(n)p Fu(Z)p Fy(.)432 3559 y Fx(D)n(\023)-54 b(e\014nition:)39 b Fy(On)34 b(dit)g(qu'un)h(group)s(e)f Fs(G)g Fy(est)g(monog)m(\022)-46 b(ene)35 b(si)f(il)g(existe)h Fs(x)30 b Fp(2)h Fs(G)i Fy(tel)i(que)432 3679 y Fs(G)27 b Fy(=)p Fs(<)h(x)g(>)p Fy(.)432 3801 y Fx(Exemple:)38 b Fy(On)33 b(a)f(mon)m(tr)m(\023)-46 b(e)33 b(que)h(les)f(sous-group)s(es)h(de)f Fu(Z)g Fy(son)m(t)g(tous)g (monog)m(\022)-46 b(enes.)432 3966 y @beginspecial @setspecial @endspecial 131 x Fw(3.2.2)136 b(PGCD)44 b(de)h(deux)g(en)l(tiers)432 4284 y Fx(D)n(\023)-54 b(e\014nition:)52 b Fy(Soit)45 b Fs(a)h Fy(et)f Fs(b)g Fy(deux)h(en)m(tiers)h(relatifs.)f(L'en)m(tier) g(p)s(ositif)f Fs(n)g Fy(tel)h(que)g Fs(<)432 4404 y(n)28 b(>)p Fy(=)p Fs(<)f Fp(f)p Fs(a;)17 b(b)p Fp(g)28 b Fs(>)i Fy(est)h(app)s(el)m(\023)-46 b(e)31 b(plus)g(grand)f(comm)m(un)i 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Fy(Consid)m(\023)-46 b(erons)25 b Fs(G)j Fy(=)f Fp(f)p Fs(au)r Fy(+)r Fs(bv)t Fy(;)17 b(\()p Fs(u;)g(v)t Fy(\))25 b Fp(2)k Fu(Z)r Fp(\002)r Fu(Z)p Fp(g)p Fy(.)23 b(Il)g(est)g(facile)g (de)g(v)m(oir)83 2138 y(que)30 b Fs(G)e Fy(est)i(un)f(sous-group)s(e)g (de)g Fu(Z)g Fy(et)g(qu'il)g(con)m(tien)m(t)h(\022)-49 b(a)29 b(la)f(fois)h Fs(a)g Fy(et)g Fs(b)p Fy(.)g(P)m(ar)g(d)m(\023)-46 b(e\014nition)83 2259 y Fs(<)28 b(a;)17 b(b)28 b(>)j Fy(est)i(l'in)m(tersection)h(de)e(tous)g(les)g(sous-group)s(es)h(de)f Fu(Z)g Fy(qui)h(con)m(tiennen)m(t)h Fs(a)e Fy(et)83 2379 y Fs(b)p Fy(.)e(Donc)g Fs(<)e(a;)17 b(b)28 b(>)p Fp(\032)g Fs(G)p Fy(,)i(soit)g(\()p Fs(a)17 b Fp(^)g Fs(b)p Fy(\))p Fu(Z)28 b Fy(=)p Fs(<)g(a)16 b Fp(^)h Fs(b)29 b(>)p Fp(\032)f Fs(G)p Fy(.)i(R)m(\023)-46 b(ecipro)s(quemen)m(t,)32 b(comme)83 2499 y Fs(<)40 b(a;)17 b(b)41 b(>)f Fy(con)m(tien)m(t)h Fs(a)f Fy(et)h Fs(b)p Fy(,)f(il)g(est)e(\023)-46 b(eviden)m(t)42 b(qu'il)f(en)g(con)m(tien)m(t)g(les)g(com)m(binaisons)h(:)83 2620 y Fs(G)28 b Fp(\032)p Fs(<)g(a;)17 b(b)28 b(>)p Fy(=)f(\()p Fs(a)22 b Fp(^)h Fs(b)p Fy(\))p Fu(Z)p Fy(.)p 3250 2620 4 66 v 3254 2557 59 4 v 3254 2620 V 3312 2620 4 66 v 83 2781 a @beginspecial @setspecial @endspecial 80 x Fx(Corollaire)39 b(3)f(\(Iden)m(tit)n(\023)-54 b(e)37 b(de)i(Bezout\).)i Fz(Soit)35 b Fs(a)h Fz(et)g Fs(b)g Fz(deux)f(entiers)g(r)-5 b(elatifs.)35 b Fs(a)h Fz(et)83 2982 y Fs(b)g Fz(sont)f(pr)-5 b(emiers)34 b(entr)-5 b(e)36 b(eux)f(si)g(et)g(seulement)g(si)g(il)g(existe)g(des)g(entiers)g(r)-5 b(elatifs)35 b Fs(u)g Fz(et)83 3102 y Fs(v)k Fz(tels)34 b(que)h Fs(au)22 b Fy(+)g Fs(bv)32 b Fy(=)c(1)p Fs(:)83 3344 y Fz(D)n(\023)-47 b(emonstr)-5 b(ation.)46 b Fy(Supp)s(osons)39 b(qu'il)f(existe)h(existe)g(des)f(en)m(tiers)h(relatifs)f Fs(u)f Fy(et)g Fs(v)k Fy(tels)83 3464 y(que)i Fs(au)29 b Fy(+)f Fs(bv)49 b Fy(=)43 b(1)p Fs(:)f(a)29 b Fp(^)h Fs(b)42 b Fy(divise)i Fs(a)e Fy(et)h Fs(b)p Fy(,)g(donc)f(divise)i Fs(au)29 b Fy(+)f Fs(bv)48 b Fy(=)c(1)p Fs(:)e Fy(Comme)i(il)83 3585 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100 x Fx(Corollaire)k(4.)k Fz(Si)34 b Fs(a)23 b Fp(^)f Fs(n)28 b Fy(=)g(1)34 b Fz(et)h Fs(b)23 b Fp(^)f Fs(n)28 b Fy(=)g(1)p Fz(,)34 b(alors)h Fy(\()p Fs(ab)p Fy(\))p Fp(j)p Fs(w)s(edg)t(en)27 b Fy(=)g(1)p Fz(.)p eop end %%Page: 28 32 TeXDict begin 28 31 bop 432 291 a @beginspecial @setspecial @endspecial Fy(28)1795 b Ft(CHAPITRE)34 b(3.)65 b(DIVISIBILIT)3615 265 y(\023)3602 291 y(E)432 555 y Fz(D)n(\023)-47 b(emonstr)-5 b(ation.)46 b Fy(\()p Fs(ab)p Fy(\))31 b Fp(^)g Fs(n)45 b Fy(est)g(un)h(diviseur)g(de)f Fs(n)p Fy(.)g(Il)g(est)h(donc)f (premier)h(a)m(v)m(ec)g Fs(a)432 676 y Fy(puisque)32 b Fs(a)e Fy(est)g(premier)h(a)m(v)m(ec)h Fs(n)p Fy(.)e(Mais)g(\()p Fs(ab)p Fy(\))18 b Fp(^)f Fs(n)30 b Fy(est)g(aussi)h(un)g(diviseur)g (de)g Fs(ab)p Fy(.)f(C'est)432 796 y(donc)j(un)g(diviseur)h(de)g Fs(b)p Fy(,)f(d'apr)m(\022)-46 b(es)34 b(de)f(lemme)h(de)f(Gauss.)g (C'est)h(un)f(diviseur)i Fs(b)e Fy(et)g(de)432 916 y Fs(n)p Fy(,)g(donc)g(c'est)g(un)g(diviseur)i(de)e Fs(b)22 b Fp(^)h Fs(n)28 b Fy(=)f(1)33 b(:)f(c'est)i(donc)f(1.)p 3599 916 4 66 v 3603 854 59 4 v 3603 916 V 3661 916 4 66 v 578 1145 a(Remarque)43 b(:)e(cette)i(prop)s(osition)f(p)s(eut)g (s')m(\023)-46 b(etendre)43 b(par)e(r)m(\023)-46 b(ecurrence)44 b(au)e(pro)s(duit)432 1265 y(d'un)33 b(nom)m(bre)g(quelconque)i(de)f (termes.)432 1389 y Fx(R)n(\023)-54 b(esolution)37 b(de)h(l')n(\023)-54 b(equation)39 b(de)f(l')n(\023)-54 b(equation)38 b Fs(au)22 b Fy(+)g Fs(bv)32 b Fy(=)c Fs(n)578 1513 y Fy(Soien)m(t)f Fs(a;)17 b(b;)g(n)27 b Fy(en)m(tiers)h(relatifs)f(\014x)m(\023)-46 b(es)28 b(On)f(v)m(eut)h(trouv)m(er)f(tous)g(les)h(couples)g(\()p Fs(u;)17 b(v)t Fy(\))26 b Fp(2)432 1634 y Fu(Z)f Fp(\002)g Fu(Z)36 b Fy(tels)h(que)g Fs(au)24 b Fy(+)g Fs(bv)38 b Fy(=)33 b Fs(n)p Fy(.)j(T)-8 b(out)36 b(d'ab)s(ord,)g(on)g(a)g(vu)g (au)g(th)m(\023)-46 b(eor)m(\022)g(eme)37 b @beginspecial @setspecial @endspecial(10)p @beginspecial @setspecial @endspecial 36 w(que)g(les)432 1754 y(en)m(tiers)31 b(de)g(la)f(forme)g Fs(au)17 b Fy(+)g Fs(bv)32 b Fy(\023)-46 b(etaien)m(t)30 b(exactemen)m(t)j(les)e(m)m(ultiples)h(de)f Fs(a)17 b Fp(^)g Fs(b)p Fy(.)31 b(Donc)f(si)432 1874 y Fs(a)22 b Fp(^)h Fs(b)33 b Fy(ne)g(divise)h(pas)f Fs(n)p Fy(,)g(il)f(n'y)i(a)e (pas)h(de)g(solution.)578 1999 y(Sinon,)46 b(consid)m(\023)-46 b(erons)47 b(une)e(solution)h(particuli)m(\022)-46 b(ere)46 b(\()p Fs(u)2622 2014 y Fm(0)2661 1999 y Fs(;)17 b(v)2752 2014 y Fm(0)2791 1999 y Fy(\))45 b(\(une)h(telle)g(solution)432 2119 y(p)s(eut)38 b(^)-46 b(etre)41 b(trouv)m(\023)-46 b(ee)41 b(en)g(appliquan)m(t)h(la)e(m)m(\023)-46 b(etho)s(de)41 b(d'Euclide,)i(puis)e(en)g(m)m(ultiplian)m(t)432 2239 y(par)658 2200 y Fn(n)p 622 2216 115 4 v 622 2274 a(a)p Fo(^)p Fn(b)785 2239 y Fy(la)e(solution)h(trouv)m(\023)-46 b(ee.\))40 b(Soit)f(donc)h(\()p Fs(u)2238 2254 y Fm(0)2277 2239 y Fs(;)17 b(v)2368 2254 y Fm(0)2407 2239 y Fy(\))39 b(tel)h(que)g Fs(au)2927 2254 y Fm(0)2993 2239 y Fy(+)26 b Fs(bv)3183 2254 y Fm(0)3262 2239 y Fy(=)39 b Fs(n)p Fy(.)h(Soit)432 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Fy(\).)30 b(Mais)h Fs(b)2150 3229 y Fo(0)2204 3265 y Fy(et)f Fs(a)2366 3229 y Fo(0)2419 3265 y Fy(son)m(t)h(premiers)g(en)m(tre)g(eux,)g(donc)432 3385 y Fs(b)473 3349 y Fo(0)528 3385 y Fy(divise)i Fs(u)20 b Fp(\000)h Fs(u)1031 3400 y Fm(0)1101 3385 y Fy(:)32 b(il)g(existe)h(un)f(en)m(tier)h(relatif)f Fs(k)i Fy(tel)e(que)h Fs(u)20 b Fp(\000)g Fs(u)2851 3400 y Fm(0)2918 3385 y Fy(=)27 b Fs(k)s(b)3116 3349 y Fo(0)3140 3385 y Fy(.)32 b(On)g(conclut)432 3506 y(alors)g(que)i Fs(v)26 b Fp(\000)c Fs(v)1066 3521 y Fm(0)1134 3506 y Fy(=)27 b Fp(\000)p Fs(k)s(a)1419 3469 y Fo(0)1443 3506 y Fy(.)33 b(Ainsi,)g(toute)g (solution)g(s')m(\023)-46 b(ecrit)34 b(sous)f(la)g(forme)1453 3761 y(\()p Fs(u;)17 b(v)t Fy(\))26 b(=)i(\()p Fs(u)1904 3776 y Fm(0)1943 3761 y Fs(;)17 b(v)2034 3776 y Fm(0)2073 3761 y Fy(\))22 b(+)g Fs(k)s Fy(\()p Fs(b)2364 3720 y Fo(0)2388 3761 y Fs(;)17 b Fp(\000)p Fs(a)2560 3720 y Fo(0)2584 3761 y Fy(\))p Fs(;)432 3948 y Fy(o)s(\022)-51 b(u)26 b(\()p Fs(u)656 3963 y Fm(0)695 3948 y Fs(;)17 b(v)786 3963 y Fm(0)825 3948 y Fy(\))27 b(est)g(une)h(solution)g (particuli)m(\022)-46 b(ere.)28 b(On)f(v)m(\023)-46 b(eri\014e)28 b(facilemen)m(t)h(que)f(tout)f(couple)432 4069 y(qui)33 b(s')m(\023)-46 b(ecrit)34 b(ainsi)f(est)g(une)g(solution)h(de)f(l')m (\023)-46 b(equation)33 b Fs(au)22 b Fy(+)g Fs(bv)32 b Fy(=)27 b Fs(n)p Fy(.)432 4193 y Fx(Exemple:)38 b Fy(R)m(\023)-46 b(esolution)33 b(de)g(l')m(\023)-46 b(equation)34 b(30)p Fs(u)21 b Fy(+)h(12)p Fs(v)31 b Fy(=)d(24.)578 4317 y(On)33 b(applique)h(l'Algorithme)f(d'Euclide)p 749 4563 2603 4 v 747 4684 4 121 v 791 4648 a Fs(n)p 889 4684 V 906 4684 V 100 w(a)1000 4663 y Fn(n)p 1087 4684 V 1135 4648 a Fs(b)1176 4663 y Fn(n)p 1267 4684 V 1311 4648 a Fs(q)1354 4663 y Fn(n)p 1441 4684 V 1530 4648 a Fs(a)p Fy(=)56 b Fs(b)p Fp(\002)s Fs(q)t Fy(+)r Fs(r)p 2046 4684 V 2063 4684 V 102 w(u)2161 4663 y Fn(n)p 2248 4684 V 2307 4648 a Fs(v)2354 4663 y Fn(n)2450 4648 y Fy(=)32 b Fs(u)2614 4663 y Fn(n)p Fm(+1)2783 4648 y Fp(\000)h Fs(q)2936 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(divise)j(bien)e(24,)f(donc)h(il)g(y)g(a)f(une)h(solution.)g(Gr^)-49 b(ace)31 b(\022)-49 b(a)30 b(l'algorithme)432 5370 y(d'Euclide,)39 b(on)f(obtien)m(t)g(la)f(solution)i(de)f(l')m(\023)-46 b(equation)38 b(de)g(Bezout)g(asso)s(ci)m(\023)-46 b(ee)39 b(30)25 b Fp(\002)h Fy(1)g(+)p eop end %%Page: 29 33 TeXDict begin 29 32 bop 83 291 a @beginspecial @setspecial @endspecial Ft(3.2.)65 b(PGCD)2635 b Fy(29)83 555 y(12)32 b Fp(\002)h Fy(\()p Fp(\000)p Fy(2\))53 b(=)g(6)48 b(En)g(m)m (ultiplian)m(t)h(par)e(4,)h(cela)g(nous)g(donne)h(bien)f(une)g (solution)83 676 y(particuli)m(\022)-46 b(ere)34 b(:)e(\()p Fs(u)751 691 y Fm(0)790 676 y Fs(;)17 b(v)881 691 y Fm(0)921 676 y Fy(\))27 b(=)h(\(4)p Fs(;)17 b Fp(\000)p Fy(8\))32 b(:)1169 843 y(30)22 b Fp(\002)g Fy(4)g(+)g(12)g Fp(\002)h Fy(\()p Fp(\000)p Fy(8\))28 b(=)f(24)p Fs(:)229 1010 y Fy(On)33 b(a)f(30)p Fs(=)p Fy(\(30)21 b Fp(^)i Fy(12\))k(=)g(30)p Fs(=)p Fy(6)g(=)h(5)k(tandis)h(que)h(12)p Fs(=)p Fy(\(30)21 b Fp(^)h Fy(12\))27 b(=)h(12)p Fs(=)p 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952 1444 4 v 978 1072 4 121 v 1026 1036 a(2)p 1120 1072 V 1137 1072 V 129 w(10)p 1366 1072 V 131 w(5)p 1546 1072 V 128 w(2)p 1720 1072 V 153 w(10=)g(5)p Fp(\002)p Fy(2+)g(0)p 2422 1072 V 980 1075 1444 4 v 978 1196 4 121 v 1026 1160 a(3)p 1120 1196 V 1137 1196 V 153 w(5)p 1366 1196 V 156 w(0)p 1546 1196 V 108 w(PGCD)27 b(=)h(5)p 2422 1196 V 980 1199 1444 4 v 229 1364 a(On)33 b(a)f(alors)h(120)21 b Fp(_)i Fy(55)k(=)1203 1325 y Fm(120)p Fo(\002)p Fm(55)p 1203 1341 232 4 v 1207 1398 a(120)p Fo(^)p Fm(55)1471 1364 y Fy(=)1585 1325 y Fm(120)p Fo(\002)p Fm(55)p 1585 1341 V 1683 1398 a(5)1854 1364 y Fy(=)h(120)21 b Fp(\002)i Fy(11)k(=)g(1320)p Fs(:)83 1572 y @beginspecial @setspecial @endspecial 153 x Fq(3.4)161 b(Congruence)83 1953 y Fx(D)n(\023)-54 b(e\014nition:)41 b Fy(Soit)35 b Fs(n)g Fy(un)h(en)m(tier)g(non)f(n)m (ul.)h(On)f(d)m(\023)-46 b(e\014nit)36 b(une)g(relation)f(binaire)h (sur)f Fu(Z)83 2074 y Fy(app)s(el)m(\023)-46 b(ee)34 b(congruence)g(mo)s(dulo)e Fs(n)h Fy(et)g(not)m(\023)-46 b(ee)33 b Fs(:)28 b Fp(\021)g Fs(:)p Fy([)p Fs(n)p Fy(])33 b(d)m(\023)-46 b(e\014nie)34 b(par)1050 2333 y(\()p Fs(a)28 b Fp(\021)g Fs(b)33 b Fy([)p Fs(n)p Fy(]\))56 b Fp(\()-17 b(\))55 b Fy(\()p Fs(a)22 b Fp(\000)h Fs(b)28 b Fp(2)g Fs(n)p Fu(Z)p Fy(\))p Fs(:)229 2528 y Fy(Il)22 b(est)h(facile)f(de)g(v) m(oir)g(que)h(la)f(congruence)h(mo)s(dulo)f Fs(n)f Fy(est)i(une)f (relation)g(d')m(\023)-46 b(equiv)-5 b(alence.)229 2653 y(Quelques)35 b(propri)m(\023)-46 b(et)m(\023)g(es)34 b(:)83 2699 y @beginspecial @setspecial @endspecial 202 2822 a(1.)49 b Fs(a)28 b Fp(\021)g Fs(b)33 b Fy([)p Fs(n)p Fy(])g(et)g Fs(c)27 b Fp(\021)h Fs(d)33 b Fy([)p Fs(n)p Fy(])g(impliquen)m(t)h Fs(a)23 b Fy(+)f Fs(c)27 b Fp(\021)i Fs(b)22 b Fy(+)g Fs(d)32 b Fy([)p Fs(n)p Fy(])83 2868 y @beginspecial @setspecial @endspecial 202 2986 a(2.)49 b Fs(a)28 b Fp(\021)g Fs(b)33 b Fy([)p Fs(n)p Fy(])g(et)g Fs(c)27 b Fp(\021)h Fs(d)33 b Fy([)p Fs(n)p Fy(])g(impliquen)m(t)h Fs(ac)28 b Fp(\021)g Fs(bd)33 b Fy([)p Fs(n)p Fy(])83 3033 y @beginspecial @setspecial @endspecial 202 3150 a(3.)49 b Fs(a)28 b Fp(\021)g Fs(b)33 b Fy([)p Fs(n)p Fy(])g(implique)h Fp(8)p Fs(k)d Fp(2)d Fu(N)98 b Fs(a)1580 3114 y Fn(k)1650 3150 y Fp(\021)29 b Fs(b)1797 3114 y Fn(k)1872 3150 y Fy([)p Fs(n)p Fy(].)83 3197 y @beginspecial @setspecial @endspecial 202 3314 a(4.)49 b Fs(a)28 b Fp(\021)g Fs(b)33 b Fy([)p Fs(n)p Fy(])g(implique)h Fs(ac)28 b Fp(\021)g Fs(bc)33 b Fy([)p Fs(nc)p Fy(])83 3361 y @beginspecial @setspecial @endspecial 202 3478 a(5.)49 b(Si)33 b Fs(d)f Fy(divise)i Fs(n)p Fy(,)f(alors)g Fs(a)27 b Fp(\021)i Fs(b)k Fy([)p Fs(n)p Fy(])g(implique)h Fs(a)28 b Fp(\021)g Fs(b)33 b Fy([)p Fs(d)p Fy(])83 3525 y @beginspecial @setspecial @endspecial 202 3642 a(6.)49 b(Si)33 b Fs(a)28 b Fp(\021)g Fs(b)33 b Fy([)p Fs(n)p Fy(])g(et)f Fs(a)c Fp(\021)h Fs(b)k Fy([)p Fs(m)p Fy(],)g(alors)f Fs(a)c Fp(\021)g Fs(b)33 b Fy([)p Fs(n)23 b Fp(_)f Fs(m)p Fy(])83 3811 y Fx(Exemple:)38 b Fy(Calcul)c(du)f(c)m(hi\013re)g(des)h(unit)m (\023)-46 b(es)34 b(de)f(l')m(\023)-46 b(ecriture)34 b(d)m(\023)-46 b(ecimale)34 b(de)f(3)2889 3775 y Fm(402)2999 3811 y Fy(.)f(On)h(a)1197 4037 y(3)1246 4000 y Fm(2)1313 4037 y Fp(\021)28 b(\000)p Fy(1)33 b([10])1197 4157 y(3)1246 4121 y Fm(4)1313 4157 y Fp(\021)28 b Fy(1)33 b([10])1197 4277 y(3)1246 4241 y Fm(400)1384 4277 y Fp(\021)28 b Fy(1)k([10])1197 4398 y(3)1246 4362 y Fm(402)1384 4398 y Fp(\021)c Fy(3)1538 4362 y Fm(400)1648 4398 y Fs(:)p Fy(3)1724 4362 y Fm(2)1791 4398 y Fp(\021)g(\000)p Fy(1)33 b([10])1197 4518 y(3)1246 4482 y Fm(402)1384 4518 y Fp(\021)28 b Fy(9)k([10])229 4759 y(Donc)h(le)g(c)m(hi\013re)g(des)h(unit)m(\023) -46 b(es)34 b(de)f(l')m(\023)-46 b(ecriture)34 b(d)m(\023)-46 b(ecimale)34 b(de)f(3)2457 4723 y Fm(402)2599 4759 y Fy(est)g(9.)83 4884 y Fx(Exemple:)38 b Fy(En)33 b(utilisan)m(t)h(les)f (congruences)i(mo)s(dulo)d(5,)h(on)f(mon)m(tre)h(que)h(l')m(\023)-46 b(equation)83 5005 y Fs(x)138 4969 y Fm(3)200 5005 y Fp(\000)23 b Fy(13)p Fs(x)f Fy(+)g(6)27 b(=)h(0)k(n'a)h(pas)g(de)g (solution)g(en)m(ti)m(\022)-46 b(ere.)229 5130 y(On)25 b(remarque)h(d'ab)s(ord)e(que)i Fs(x)1388 5094 y Fm(3)1434 5130 y Fp(\000)6 b Fy(13)p Fs(x)g Fy(+)g(6)27 b Fp(\021)h Fs(x)1994 5094 y Fm(3)2040 5130 y Fy(+)6 b(2)p Fs(x)g Fy(+)g(1)24 b([5])g(\(La)h(nouv)m(elle)h(expre-)83 5250 y(sion)36 b(est)f(plus)h(facile)g(\022)-49 b(a)34 b(calculer\).)j (Ensuite)f(en)g(teste)g(p)s(our)e(toutes)i(les)g(congruences)83 5370 y(mo)s(dulo)d(5)f(p)s(ossibles)p eop end %%Page: 32 36 TeXDict begin 32 35 bop 432 291 a @beginspecial @setspecial @endspecial Fy(32)1795 b Ft(CHAPITRE)34 b(3.)65 b(DIVISIBILIT)3615 265 y(\023)3602 291 y(E)1285 762 y Fs(x)29 b Fp(\021)f Fy(0)k([5])83 b(=)-17 b Fp(\))28 b Fs(x)1983 726 y Fm(3)2045 762 y Fy(+)22 b(2)p Fs(x)g Fy(+)g(1)27 b Fp(\021)i Fy(1)j([5])1285 908 y Fs(x)d Fp(\021)f Fy(1)k([5])83 b(=)-17 b Fp(\))28 b Fs(x)1983 871 y Fm(3)2045 908 y Fy(+)22 b(2)p Fs(x)g Fy(+)g(1)27 b Fp(\021)i Fy(3)j([5])1285 1053 y Fs(x)d Fp(\021)f Fy(2)k([5])83 b(=)-17 b Fp(\))28 b Fs(x)1983 1017 y Fm(3)2045 1053 y Fy(+)22 b(2)p Fs(x)g Fy(+)g(1)27 b Fp(\021)i Fy(3)j([5])1285 1198 y Fs(x)d Fp(\021)f Fy(3)k([5])83 b(=)-17 b Fp(\))28 b Fs(x)1983 1162 y Fm(3)2045 1198 y Fy(+)22 b(2)p Fs(x)g Fy(+)g(1)27 b Fp(\021)i Fy(4)j([5])1285 1343 y Fs(x)d Fp(\021)f Fy(4)k([5])83 b(=)-17 b Fp(\))28 b Fs(x)1983 1307 y Fm(3)2045 1343 y Fy(+)22 b(2)p Fs(x)g Fy(+)g(1)27 b Fp(\021)i Fy(3)j([5])578 1696 y(Ainsi,)g(p)s(our)e(tout)h Fs(x)g Fy(en)m(tier,)h Fs(x)1734 1660 y Fm(3)1792 1696 y Fp(\000)19 b Fy(13)p Fs(x)f Fy(+)g(6)31 b(n'est)g(jamais)g(congru)g (\022)-49 b(a)31 b(0)f(mo)s(dulo)h(5.)432 1816 y(Il)i(ne)g(p)s(eut)f (donc)i(jamais)c(^)-46 b(etre)32 b(n)m(ul.)432 1984 y @beginspecial @setspecial @endspecial 163 x Fq(3.5)160 b(Exercices)432 2230 y @beginspecial @setspecial @endspecial 551 2366 a Fy(1.)49 b(Mon)m(trer)31 b(qu'un)g(en)m(tier)g(natuel)g(est) g(congrus)g(\022)-49 b(a)30 b(la)g(somme)h(des)h(c)m(hi\013res)f(de)g (son)673 2486 y(\023)-46 b(ecriture)33 b(dans)g(la)g(base)g Fs(b)g Fy(mo)s(dulo)g Fs(b)22 b Fp(\000)h Fy(1.)432 2510 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 551 2637 a(2.)90 b(\(a\))899 2612 y(\023)890 2637 y(Etudier,)46 b(suiv)-5 b(an)m(t)47 b(les)f(v)-5 b(aleurs)45 b(de)h(l'en)m(tier)h(naturel)e Fs(n)p Fy(,)g(le)h(reste)g(de)g(la)890 2758 y(division)34 b(par)f(7)f(du)h(nom)m(bre)g Fs(A)28 b Fy(=)g Fs(n)2259 2722 y Fm(2)2320 2758 y Fp(\000)23 b Fs(n)f Fy(+)g(1.)432 2792 y @beginspecial @setspecial @endspecial 711 2909 a(\(b\))49 b(En)33 b(d)m(\023)-46 b(eduire)34 b(les)f(en)m(tiers)i Fs(n)d Fy(tels)i(que)f(le)g(nom)m(bre)g Fs(A)g Fy(soit)g(divisible)h (par)f(7.)432 2949 y @beginspecial @setspecial @endspecial 722 3059 a(\(c\))49 b(D)m(\023)-46 b(eterminer)37 b(le)e(reste)i(de)f (la)f(division)i(par)e(7)g(du)h(nom)m(bre)g Fs(B)i Fy(=)32 b(2753)3529 3023 y Fm(2)3592 3059 y Fp(\000)890 3180 y Fy(2753)21 b(+)h(1.)676 3315 y Fz(Bac)-5 b(c)g(alaur)n(\023)-47 b(eat)33 b(s)n(\023)-47 b(erie)33 b(C)h(1968,)h(Camb)-5 b(o)g(dge)33 b(et)i(L)-5 b(aos)432 3350 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 551 3466 a Fy(3.)90 b(\(a\))48 b(D)m(\023)-46 b(eterminer,)34 b(dans)f(l'ensem)m(ble)i Fu(N)e Fy(des)g(en)m(tiers)i(naturels,)e (toutes)g(les)h(so-)890 3587 y(lutions)f(de)h(l')m(\023)-46 b(equation)2013 3707 y(2)p Fs(x)23 b Fp(\000)f Fy(3)p Fs(y)31 b Fy(=)d(0)p Fs(:)432 3775 y @beginspecial @setspecial @endspecial 711 3891 a Fy(\(b\))49 b(D)m(\023)-46 b(eterminer,)38 b(dans)f(l'ensem)m(ble)j Fu(N)d Fy(des)g(en)m(tiers)i(naturels,)f(une)f (solution)890 4012 y(de)c(l')m(\023)-46 b(equation)2013 4132 y(2)p Fs(x)23 b Fp(\000)f Fy(3)p Fs(y)31 b Fy(=)d(3)p Fs(:)890 4301 y Fy(En)33 b(d)m(\023)-46 b(eduire)34 b(toutes)f(les)h (autres)f(solutions.)676 4436 y Fz(Bac)-5 b(c)g(alaur)n(\023)-47 b(eat)33 b(s)n(\023)-47 b(erie)33 b(C,)h(septembr)-5 b(e)34 b(1968,)g(Montr)n(\023)-47 b(eal)34 b(et)h(New)g(Y)-7 b(ork)432 4471 y @beginspecial @setspecial @endspecial 551 4587 a Fy(4.)49 b(T)-8 b(rouv)m(er)37 b(le)f(reste)i(dans)e(la)g (division)i(par)d(7)h(du)h(nom)m(bre)g(65)2954 4551 y Fm(346)3063 4587 y Fy(.)g Fz(Bac)-5 b(c)g(alaur)n(\023)-47 b(eat)676 4708 y(s)n(\023)g(erie)33 b(Math)n(\023)-47 b(ematiques)31 b(\023)-47 b(el)n(\023)g(ementair)-5 b(es,)32 b(1967,)i(A)n(ntil)5 b(les)432 4742 y @beginspecial @setspecial @endspecial 551 4858 a Fy(5.)49 b(Construire)37 b(tous)f(les)h(couples) g(d'en)m(tiers)h(p)s(ositifs)e(v)m(\023)-46 b(eri\014an)m(t)37 b(la)f(relation)g(3)p Fs(x)25 b Fy(+)676 4979 y(13)p Fs(y)30 b Fy(=)d(166.)432 5013 y @beginspecial @setspecial @endspecial 551 5130 a(6.)49 b(Quels)35 b(son)m(t)g(les)h(en)m(tiers)g (p)s(ositifs)f Fs(n)g Fy(p)s(our)f(lesquels)j(15)23 b Fp(\002)h Fy(3)2911 5094 y Fn(n)2981 5130 y Fp(\000)g Fy(3)34 b(est)h(divisible)676 5250 y(par)28 b(7)16 b(?)28 b Fz(Bac)-5 b(c)g(alaur)n(\023)-47 b(eat)29 b(s)n(\023)-47 b(erie)30 b(math.)e(\023)-47 b(elem.,)29 b(session)h(o)-5 b(ctobr)g(e-novembr)g(e)29 b(1967,)676 5370 y(Mexic)-5 b(o)p eop end %%Page: 33 37 TeXDict begin 33 36 bop 83 291 a @beginspecial @setspecial @endspecial Ft(3.5.)65 b(EXER)m(CICES)2354 b Fy(33)83 456 y @beginspecial @setspecial @endspecial 202 555 a(7.)49 b(Quel)32 b(p)s(eut)d(^)-46 b(etre,)31 b(suiv)-5 b(an)m(t)33 b(les)f(v)-5 b(aleurs)32 b(du)g(nom)m(bre)g(en)m(tier)h(naturel)e Fs(n)p Fy(,)h(le)g(reste)327 676 y(de)37 b(la)f(division)i(par)f(7)f (du)h(nom)m(bre)g(2)1748 639 y Fn(n)1811 676 y Fy(?)g(En)g(d)m(\023)-46 b(eduire)38 b(le)f(reste)g(de)g(la)f(division)327 796 y(par)c(7)h(du)g(nom)m(bre)g Fs(N)38 b Fy(=)28 b(247)1439 760 y Fm(349)1548 796 y Fy(.)83 832 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 202 951 a(8.)90 b(\(a\))49 b(Sans)44 b(e\013ectuer)i(la)e(division)h(de)g (761945)e(par)h(11,)g(trouv)m(er)h(le)f(reste)h(de)542 1071 y(cette)33 b(op)m(\023)-46 b(eration.)83 1108 y @beginspecial @setspecial @endspecial 363 1226 a(\(b\))49 b(Mon)m(trer)34 b(qu'en)g(c)m(hangean)m(t)g(un)g(seul)g(c)m(hi\013re)h (du)e(dividende,)j(on)d(p)s(eut)g(le)542 1346 y(rendre)27 b(divisible)h(par)f(11.)f(Indiquer)i(tous)f(les)g(nom)m(bres)h (divisibles)h(par)d(11)542 1467 y(que)33 b(l'on)g(p)s(eut)h(obtenir)f (de)g(cette)h(fa\030)-43 b(con.)33 b(Quels)h(son)m(t,)f(parmi)h(eux,)f (ceux)542 1587 y(qui)g(son)m(t)d(\023)-46 b(egalemen)m(t)34 b(divisibles)h(par)d(15)16 b(?)83 1623 y @beginspecial @setspecial @endspecial 202 1742 a(9.)49 b(D)m(\023)-46 b(eterminer)24 b(les)g(c)m(hi\013res)h Fs(x)f Fy(et)f Fs(y)k Fy(de)d(fa\030)-43 b(con)23 b(que)h(le)g(nom)m(bre)g(qui)g(dans) g(le)g(syst)m(\022)-46 b(eme)327 1862 y(d)m(\023)g(ecimal)34 b(s')m(\023)-46 b(ecrit)33 b(28)p Fs(x)p Fy(75)p Fs(y)j Fy(soit)c(divisible)j(\022)-49 b(a)32 b(la)h(fois)f(par)h(3)f(et)h(par) f(11.)83 1898 y @beginspecial @setspecial @endspecial 153 2017 a(10.)49 b(Soit)27 b Fs(q)k Fy(et)c Fs(r)j Fy(le)e(quotien)m (t)g(et)f(le)h(reste)g(de)g(la)f(division)h(d'un)g(nom)m(bre)g(en)m (tier)h Fs(a)e Fy(par)327 2137 y(un)k(nom)m(bre)h(en)m(tier)g Fs(b)p Fy(.)f(Sac)m(han)m(t)h(que)g Fs(a)19 b Fy(+)f Fs(b)h Fy(+)g Fs(r)30 b Fy(=)d(3025)j(et)h Fs(q)h Fy(=)27 b(50,)k(r)m(\023)-46 b(etablir)31 b(la)327 2257 y(division.)83 2275 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 153 2412 a(11.)90 b(\(a\))49 b(Commen)m(t)28 b(faut-il)e(c)m(hoisir)h(l'en)m(tier)h(naturel)f Fs(n)g Fy(p)s(our)f(que)i(le)f(nom)m(bre)g Fs(A)h Fy(=)542 2533 y(2)591 2496 y Fn(n)659 2533 y Fp(\000)23 b Fy(1)32 b(soit)h(divisible)i(par)d(9)16 b(?)83 2569 y @beginspecial @setspecial @endspecial 363 2687 a(\(b\))49 b(Mon)m(trer)36 b(que)g(si)g(cette)g(condition)g(est)g(r)m(\023)-46 b(ealis)m(\023)g (ee,)36 b(le)g(nom)m(bre)g Fs(A)g Fy(est)f(divi-)542 2808 y(sible)e(par)g(7.)f(Quel)h(est)g(le)g(reste)h(de)f(la)f(division) i(de)f Fs(A)g Fy(par)f(21)16 b(?)83 2844 y @beginspecial @setspecial @endspecial 153 2962 a(12.)49 b(En)40 b(utilisan)m(t)h(la)e (th)m(\023)-46 b(eorie)41 b(des)g(congruences,)g(d)m(\023)-46 b(eterminer)42 b(la)d(forme)h(g)m(\023)-46 b(en)m(\023)g(erale)327 3083 y(des)33 b(en)m(tiers)i(naturels)e Fs(n)g Fy(tels)g(que)h Fs(n)1692 3047 y Fm(3)1754 3083 y Fp(\000)22 b Fs(n)h Fy(+)f(1)32 b(soit)h(divisible)h(par)f(7.)p eop end %%Page: 34 38 TeXDict begin 34 37 bop 432 291 a @beginspecial @setspecial @endspecial Fy(34)1795 b Ft(CHAPITRE)34 b(3.)65 b(DIVISIBILIT)3615 265 y(\023)3602 291 y(E)p eop end %%Page: 35 39 TeXDict begin 35 38 bop 83 291 a @beginspecial @setspecial @endspecial 165 x @beginspecial @setspecial @endspecial 763 x Fv(Chapitre)77 b(4)83 1634 y(Nom)-6 b(bres)76 b(premiers)83 2007 y @beginspecial @setspecial @endspecial 142 x Fq(4.1)161 b(D)l(\023)-77 b(e\014nition)51 b(et)i(premi)l(\022)-77 b(eres)52 b(propri)l(\023)-77 b(et)l(\023)g(es)83 2368 y Fx(D)n(\023)-54 b(e\014nition:)38 b Fy(On)32 b(dit)g(qu'un)h(en)m (tier)g(naturel)g(est)f(un)h(nom)m(bre)g(premier)g(lorsqu'il)g(est)83 2489 y(di\013)m(\023)-46 b(eren)m(t)41 b(de)g(1)f(et)g(quil)h(n'est)g (pas)g(p)s(ossible)g(de)g(l')m(\023)-46 b(ecrire)41 b(comme)g(pro)s (duit)f(de)h(deux)83 2609 y(en)m(tiers)34 b(naturels)g(di\013)m(\023) -46 b(eren)m(ts)34 b(de)f(1.)-1316 b @beginspecial @setspecial @endspecial 202 x Fx(Lemme)40 b(6.)i Fz(Soit)36 b Fs(n)g Fz(un)g(entier)g(natur)-5 b(el)36 b(et)g Fs(p)g Fz(un)g(nombr)-5 b(e)35 b(pr)-5 b(emier.)35 b(Si)g Fs(p)h Fz(ne)g(divise)83 2931 y(p)-5 b(as)34 b Fs(n)p Fz(,)h(alors)g Fs(p)g Fz(et)g Fs(n)g Fz(sont)f(pr)-5 b(emiers)34 b(entr)-5 b(e)35 b(eux.)83 3133 y(D)n(\023)-47 b(emonstr)-5 b(ation.)46 b Fs(p)13 b Fp(^)g Fs(n)30 b Fy(est)e(un)h(diviseur)h(de)f Fs(p)p Fy(.)f(C'est)h(donc)g Fs(p)f Fy(o)s(\022)-51 b(u)27 b(1)h(Dans)g(le)h (premier)83 3253 y(cas,)k Fs(p)p Fp(j)p Fs(n)p Fy(,)g(dans)g(le)g (deuxi)m(\022)-46 b(eme)35 b(cas)e Fs(p)f Fy(et)h Fs(n)g Fy(son)m(t)g(premiers)h(en)m(tre)g(eux.)p 3250 3253 4 66 v 3254 3191 59 4 v 3254 3253 V 3312 3253 4 66 v 83 3353 a @beginspecial @setspecial @endspecial 102 x 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Fx(Corollaire)43 b(5.)h Fz(Si)39 b(un)g(entier)g Fs(n)d(>)f Fy(1)k Fz(n)-10 b('est)39 b(p)-5 b(as)39 b(pr)-5 b(emier,)38 b(il)h(admet)f(au)i(moins)e(un)83 5346 y(diviseur)c(pr)-5 b(emier)35 b(inf)n(\023)-47 b(erieur)33 b(ou)g(\023)-47 b(egal)33 b(\022)-50 b(a)1636 5275 y Fp(p)p 1719 5275 59 4 v 71 x Fs(n)p Fz(.)1653 5620 y Fy(35)p eop end %%Page: 36 40 TeXDict begin 36 39 bop 432 291 a @beginspecial @setspecial @endspecial Fy(36)1385 b Ft(CHAPITRE)35 b(4.)65 b(NOMBRES)33 b(PREMIERS)432 555 y Fz(D)n(\023)-47 b(emonstr)-5 b(ation.)46 b Fy(Soit)35 b Fs(p)f Fy(le)i(plus)f(p)s(etit)g(diviseur)i(de)e Fs(n)g Fy(strictemen)m(t)i(sup)m(\023)-46 b(erieur)36 b(\022)-49 b(a)35 b(1.)432 676 y(C'est)41 b(un)e(nom)m(bre)i(premier.)g (Soit)e Fs(q)44 b Fy(l'unique)d(en)m(tier)f(tel)g(que)h Fs(n)f Fy(=)f Fs(pq)t Fy(.)g Fs(q)44 b(>)39 b Fy(1)h(car)432 796 y(sinon)30 b Fs(n)f Fy(serait)h(premier.)h Fs(q)i Fy(est)d(donc)g(un)g(diviseur)h(de)f Fs(n)f Fy(strictemen)m(t)j(sup)m (\023)-46 b(erieur)31 b(\022)-49 b(a)29 b(1,)432 916 y(donc)k Fs(q)e Fp(\025)d Fs(p)p Fy(.)33 b(Cela)g(en)m(traine)h Fs(n)28 b Fy(=)f Fs(pq)32 b Fp(\025)c Fs(p)2019 880 y Fm(2)2058 916 y Fy(.)p 3599 916 4 66 v 3603 854 59 4 v 3603 916 V 3661 916 4 66 v 432 1126 a Fx(Exemple:)37 b Fy(Supp)s(osons)d(acquis)f(que)g(les)g(nom)m(bres)g(2)p Fs(;)17 b Fy(3)p Fs(;)g Fy(5)p Fs(;)g Fy(7)31 b(son)m(t)h(les)h(nom)m (bres)h(pre-)432 1247 y(miers)29 b(inf)m(\023)-46 b(erieurs)30 b(\022)-49 b(a)28 b(10.)g(97)g(n'est)i(divisible)g(ni)f(par)f(2,)g(ni)h (par)f(3,)h(ni)f(par)h(5,)f(ni)h(par)f(7.)g(Il)432 1367 y(n'y)33 b(a)f(pas)h(d'autre)g(nom)m(bre)g(premier)h Fs(p)e Fy(tel)h(que)g Fs(p)2347 1331 y Fm(2)2414 1367 y Fp(\024)28 b Fy(97.)k(Donc)h(97)f(est)h(un)g(nom)m(bre)432 1487 y(premier.)578 1609 y(Liste)g(des)h(nom)m(bres)g(premiers)g(en)m (tre)g(1)e(et)h(100)f(:)432 1730 y(2)g(3)g(5)h(7)f(11)g(13)g(17)g(19)g (23)g(29)g(31)g(37)g(41)g(43)h(47)f(53)g(59)g(61)g(67)g(71)g(73)g(79)g (83)g(89)g(97)p Fs(:)578 1852 y 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(du)e(PPCM)83 5153 y @beginspecial @setspecial @endspecial 97 x Fx(Th)n(\023)-54 b(eor)n(\022)g(eme)38 b(16.)k Fz(Soit)35 b Fs(a)g Fz(et)g Fs(b)g Fz(deux)g(entiers)f(natur)-5 b(els.)35 b(On)g(a)f(p)-5 b(our)35 b(tout)h Fs(p)f Fz(pr)-5 b(emier)229 5370 y({)49 b Fs(v)375 5385 y Fn(p)415 5370 y Fy(\()p Fs(a)22 b Fp(^)h Fs(b)p Fy(\))28 b(=)f(min)q(\()p Fs(v)1073 5385 y Fn(p)1113 5370 y Fy(\()p Fs(a)p Fy(\))p Fs(;)17 b(v)1331 5385 y Fn(p)1370 5370 y Fy(\()p Fs(b)p Fy(\)\))p Fz(.)p eop end %%Page: 38 42 TeXDict begin 38 41 bop 432 291 a @beginspecial @setspecial @endspecial Fy(38)1385 b Ft(CHAPITRE)35 b(4.)65 b(NOMBRES)33 b(PREMIERS)578 555 y Fz({)49 b Fs(v)724 570 y Fn(p)763 555 y Fy(\()p Fs(a)23 b Fp(_)f Fs(b)p Fy(\))28 b(=)g(max\()p Fs(v)1440 570 y Fn(p)1480 555 y Fy(\()p Fs(a)p Fy(\))p Fs(;)17 b(v)1698 570 y Fn(p)1738 555 y Fy(\()p Fs(b)p Fy(\)\))432 765 y Fz(D)n(\023)-47 b(emonstr)-5 b(ation.)432 933 y Fs(p)481 896 y Fm(min)o(\()p Fn(v)659 904 y Fh(p)696 896 y Fm(\()p Fn(a)p Fm(\))p 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Fx(Corollaire)38 b(6.)j Fz(Soient)35 b Fs(a)g Fz(et)g Fs(b)g Fz(deux)g(entiers)f(natur)-5 b(els.)35 b(On)g(a)1334 3035 y Fs(a)22 b Fp(^)h Fs(b)28 b Fy(=)1875 2953 y Fk(Y)1668 3140 y Fn(p)p Fm(:)p Fn(v)1758 3148 y Fh(p)1794 3140 y Fm(\()p Fn(a)p Fm(\)+)p Fn(v)1974 3148 y Fh(p)2012 3140 y Fm(\()p Fn(b)p Fm(\))p Fn(>)p Fm(0)2204 3035 y Fs(p)2253 2994 y Fm(min)o(\()p Fn(v)2431 3002 y Fh(p)2468 2994 y Fm(\()p Fn(a)p Fm(\))p Fn(;v)2613 3002 y Fh(p)2651 2994 y Fm(\()p Fn(b)p Fm(\)\))1327 3374 y Fs(a)22 b Fp(_)h Fs(b)28 b Fy(=)1868 3292 y Fk(Y)1662 3478 y Fn(p)p Fm(:)p Fn(v)1752 3486 y Fh(p)1787 3478 y Fm(\()p Fn(a)p Fm(\)+)p Fn(v)1967 3486 y Fh(p)2005 3478 y Fm(\()p Fn(b)p Fm(\))p Fn(>)p Fm(0)2197 3374 y Fs(p)2246 3333 y Fm(max\()p Fn(v)2438 3341 y Fh(p)2475 3333 y Fm(\()p Fn(a)p Fm(\))p Fn(;v)2620 3341 y Fh(p)2658 3333 y Fm(\()p Fn(b)p Fm(\)\))432 3673 y Fx(Exemple:)38 b Fy(Calcul)33 b(du)g(PPCM)h(et)f(du)g(PGCD)g(de)g(1248)e(et)i(2600)578 3794 y(On)g(a)1648 3919 y(1248)27 b(=)h(2)2024 3878 y Fm(5)2085 3919 y Fp(\002)23 b Fy(3)f Fp(\002)g Fy(13)432 4099 y(et)1629 4224 y(2600)27 b(=)g(2)2004 4183 y Fm(3)2065 4224 y Fp(\002)c Fy(5)2214 4183 y Fm(2)2275 4224 y Fp(\002)g Fy(13)578 4405 y(D'o)s(\022)-51 b(u)1442 4652 y(1248)21 b Fp(^)i Fy(2600)k(=)g(2)2123 4611 y Fm(3)2185 4652 y Fp(\002)22 b Fy(13)27 b(=)h(104)432 4832 y(et)1203 4956 y(1248)21 b Fp(_)i Fy(2600)k(=)g(2)1884 4915 y Fm(5)1945 4956 y Fp(\002)c Fy(3)f Fp(\002)h Fy(5)2265 4915 y Fm(2)2326 4956 y Fp(\002)g Fy(13)k(=)g(31200)578 5138 y(V)m(\023)-46 b(eri\014cation)33 b(:)1223 5362 y(104)22 b Fp(\002)g Fy(31200)27 b(=)g(3244800)f(=)i(1248)21 b Fp(\002)i Fy(2600)p Fs(:)p eop end %%Page: 39 43 TeXDict begin 39 42 bop 83 291 a @beginspecial @setspecial @endspecial Ft(4.3.)65 b(EXER)m(CICES)2354 b Fy(39)83 456 y @beginspecial @setspecial @endspecial 99 x Fq(4.3)161 b(Exercices)83 639 y @beginspecial @setspecial @endspecial 202 774 a Fy(1.)49 b(D)m(\023)-46 b(eterminer)33 b(l'ensem)m(ble)j(des) 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b(deux)g(nom)m(bres)g(son)m(t)g(premiers)h (en)m(tre)f(eux,)g(tout)f(nom)m(bre)h(premier)639 1972 y(qui)e(divise)i(leur)e(pro)s(duit)f(divise)i(l'un)g(et)e(est)i (premier)f(a)m(v)m(ec)h(l'autre)16 b(;)542 2092 y({)48 b(si)43 b(deux)h(nom)m(bres)h(son)m(t)e(premiers)h(en)m(tre)g(eux,)g (leur)f(somme)h(et)f(leur)639 2212 y(pro)s(duit)33 b(son)m(t)g(aussi)g (deux)h(nom)m(bres)g(premiers)g(en)m(tre)g(eux.)83 2246 y @beginspecial @setspecial @endspecial 363 2362 a(\(b\))49 b(Calculer)31 b(deux)h(nom)m(bres,)g(connaissan)m(t)h(leur)e(somme,)h (135,)e(et)h(leur)g(plus)542 2482 y(p)s(etit)h(m)m(ultiple)i(comm)m (un,)g(504.)327 2617 y Fz(Bac)-5 b(c)g(alaur)n(\023)-47 b(eat)33 b(s)n(\023)-47 b(erie)33 b(C,)i(session)e(de)i(septembr)-5 b(e)34 b(1968,)g(gr)-5 b(oup)g(e)35 b(I)83 2651 y @beginspecial @setspecial @endspecial 202 2767 a Fy(6.)49 b(Soit,)33 b(en)g(n)m(um)m(\023)-46 b(eration)33 b(en)g(base)h Fs(a)e Fy(\()p Fs(a)c(>)g Fy(3\))k(le)h(nom)m(bre)g Fs(N)39 b Fy(=)27 b(1320.)83 2804 y @beginspecial @setspecial @endspecial 368 2914 a(\(a\))49 b(Mon)m(trer)33 b(que)h(ce)g(nom)m(bre) g(est)g(divisible)h(par)e Fs(a)p Fy(,)g Fs(a)23 b Fy(+)f(1,)33 b Fs(a)23 b Fy(+)f(2)33 b(et)g(se)h(met)542 3034 y(sous)f(la)f(forme)h Fs(a)p Fy(\()p Fs(a)23 b Fy(+)f(1\)\()p Fs(a)g Fy(+)g(2\).)83 3074 y @beginspecial @setspecial @endspecial 363 3184 a(\(b\))49 b(P)m(our)33 b(quelles)h(v)-5 b(aleurs)34 b(de)f Fs(a)f Fy(est-il)h(divisible)i(par)d Fs(a)23 b Fp(\000)f Fy(1)16 b(?)83 3223 y @beginspecial @setspecial @endspecial 374 3333 a(\(c\))49 b(En)42 b(prenan)m(t)h(p)s(our)f(v)-5 b(aleur)43 b(de)g Fs(a)f Fy(la)g(plus)h(p)s(etite)g(de)g(celles)g (trouv)m(\023)-46 b(ees)44 b(\022)-49 b(a)542 3454 y(la)29 b(question)j(pr)m(\023)-46 b(ec)m(\023)g(eden)m(te,)32 b(d)m(\023)-46 b(ecomp)s(oser)32 b Fs(N)40 b Fy(en)31 b(pro)s(duit)f(de)g(facteurs)h(pre-)542 3574 y(miers.)327 3721 y Fz(Bac)-5 b(c)g(alaur)n(\023)-47 b(eat)33 b(s)n(\023)-47 b(erie)33 b(math.)f(\023)-47 b(elem.,)34 b(sujet)h(de)f(r)-5 b(emplac)g(ement)34 b(1967,)g(A)n(ntil)5 b(les)83 3755 y @beginspecial @setspecial @endspecial @beginspecial @setspecial @endspecial 202 3871 a Fy(7.)90 b(\(a\))49 b(Soit)32 b Fs(p)h Fy(un)f(nom)m(bre)i(premier.)g(Mon)m(trer)f(que)1484 4141 y Fs(v)1531 4156 y Fn(p)1571 4141 y Fy(\()p Fs(n)p Fy(!\))27 b(=)1865 4034 y Fm(+)p Fo(1)1867 4059 y Fk(X)1863 4242 y Fn(k)r Fm(=0)2008 4141 y Fy(En)m(t)q(\()2229 4074 y Fs(n)p 2212 4118 92 4 v 2212 4210 a(p)2261 4181 y Fn(k)2314 4141 y Fy(\))p Fs(:)83 4332 y @beginspecial @setspecial @endspecial 363 4430 a Fy(\(b\))49 b(En)33 b(d)m(\023)-46 b(eduire)33 b(l'in)m(\023)-46 b(egalit)m(\023)g(e)1266 4588 y Fs(n)p 1266 4632 59 4 v 1271 4724 a(p)1356 4656 y Fp(\000)23 b Fy(1)k Fs(<)h(v)1683 4671 y Fn(p)1723 4656 y Fy(\()p Fs(n)p Fy(!\))g Fp(\024)2027 4588 y Fs(n)p 2027 4632 V 2032 4724 a(p)2117 4656 y Fy(+)2368 4588 y Fs(n)p 2225 4632 345 4 v 2225 4724 a(p)p Fy(\()p Fs(p)22 b Fp(\000)h Fy(1\))2579 4656 y Fs(:)542 4927 y Fy(En)29 b(consid)m(\023)-46 b(eran)m(t)31 b(l'expression)g(\(1)15 b(+)g(1\))2006 4891 y Fm(2)p Fn(m)p Fm(+1)2197 4927 y Fy(,)29 b(mon)m(trer)2621 4843 y Fk(\200)2664 4883 y Fm(2)p Fn(m)p Fm(+1)2727 4962 y Fn(m)2852 4843 y Fk(\212)2924 4927 y Fp(\024)f Fy(4)3078 4891 y Fn(m)3144 4927 y Fy(.)h(En)542 5048 y(d)m(\023)-46 b(eduire)33 b(la)g(ma)5 b(joration)1708 5168 y Fk(Y)1518 5349 y Fn(m)p Fm(+1)p Fn(