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transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
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noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
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N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs
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scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg
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/setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{
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%%EndProcSet
%%BeginFont: CMR5
%!PS-AdobeFont-1.1: CMR5 1.00B
%%CreationDate: 1992 Feb 19 19:55:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR5 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 /.notdef put
readonly def
/FontBBox{-341 -250 1304 965}readonly def
/UniqueID 5000788 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171
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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 0 /.notdef put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueID 5087383 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE
3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B
532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470
B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B
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5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC
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AFB83D1FC65F19CECA428ABD1A709C03601CEA952E383C9C9A8F6CF3A724C485
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3F41977445FC2F541AB2D932A4CD385D47F83F58BF77AA2E636120037F99AE37
60BB5A0A975BFDE8E368D177C6826002F87DB0F4CE641357DB0D970F3B791208
431FEA112798FA296C7B045CB06569775EA4FD162FCD7CD5A7C70C17F35ABFDF
B8A41D8B37AC90A78403DF98AC01751B4C4576862910A58C255D20D38B478F40
CCB8A6EC52DDFC705E2D746321E3C6954EF07E6619B737D28A53CEB142430C16
466A4E806B312530E08170751091C56F6CFA00CFAAE5D7EED4A39B96CEFD40BA
4C700FA48174686D28806824842424E0262CF206376806A7E321D5A517FE5FE8
4590E07635FE56DA7B555A13B13F09B30E4246A839824EE55EE37DCBE43F3419
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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 0 /.notdef put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueID 5087381 def
currentdict end
currentfile eexec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6166526FEB90C30D30099FBDC1BC2F9B62EFEEC48345160804AA98F8D0AA54B7
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3A9444C8CD9158D84117299F7F2332FEB0F94E6ED8BC7AA789A3219BC2F227D3
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3A05E77F432C98738ABC73EC31EABAFCF7DE2255B85D6F31C2B5569A430B775A
9C3018188683B35AA10FAF838D140D1C4C40A977E758993125D79B35945CCCEB
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B8D0EEBC897B8AA811676A336BEF23EA6F4532E49DB755E21F8CCE24BB91AF9F
3BB417926EE8E47D53ADFB545C2F7C0C87B631385D95896A4C791EE9C686D680
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cleartomark
%%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 0 /.notdef put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171
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cleartomark
%%EndFont
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 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 20 /bracketleftbigg put
dup 21 /bracketrightbigg put
dup 112 /radicalbig put
dup 114 /radicalbigg put
dup 115 /radicalBigg put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueID 5000774 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53
% 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 (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 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 /.notdef put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueID 5087382 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52
% 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 (CMSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY7 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 /.notdef put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 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 3 /square put
readonly def
/FontBBox{8 -463 1331 1003}readonly def
/UniqueID 5031981 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
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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 /.notdef put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% 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 (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 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 /.notdef put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 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 /.notdef put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
/UniqueID 5000768 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR7
%!PS-AdobeFont-1.1: CMR7 1.0
%%CreationDate: 1991 Aug 20 16:39:21
% 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 (CMR7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR7 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 /.notdef put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
/UniqueID 5000790 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMTT10
%!PS-AdobeFont-1.1: CMTT10 1.00B
%%CreationDate: 1992 Apr 26 10:42:42
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTT10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT10 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 /.notdef put
readonly def
/FontBBox{-4 -235 731 800}readonly def
/UniqueID 5000832 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 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 /.notdef put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 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 /.notdef put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueID 5000828 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMCSC10
%!PS-AdobeFont-1.1: CMCSC10 1.0
%%CreationDate: 1991 Aug 18 17:46:49
% 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 (CMCSC10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMCSC10 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 /.notdef put
readonly def
/FontBBox{14 -250 1077 750}readonly def
/UniqueID 5000772 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
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end readonly def
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readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueID 5000794 def
currentdict end
currentfile eexec
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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 0 /.notdef put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueID 5000769 def
currentdict end
currentfile eexec
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b(1)18 b(+)g(13)f(+)h(17)g(+)g(19)27 b(\(mo)r(d)h(25\))1714
2746 y(=)23 b(0)k(\(mo)r(d)h(25\))456 2924 y(as)k(desired.)51
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3023 y(random,)d(but)h(\\magically")d(add)i(up)h(to)f(the)h(righ)n(t)e
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y(1)p 817 3315 V 817 3392 a(1)887 3335 y(+)980 3278 y(1)p
980 3315 V 980 3392 a(2)1050 3335 y(+)1143 3278 y(1)p
1143 3315 V 1143 3392 a(3)1213 3335 y(+)1306 3278 y(1)p
1306 3315 V 1306 3392 a(4)1376 3335 y(+)1469 3278 y(1)p
1469 3315 V 1469 3392 a(5)1539 3335 y(+)1632 3278 y(1)p
1632 3315 V 1632 3392 a(6)1707 3335 y(=)g(1)c(+)g(25)g(+)g(33)f(+)h(37)
g(+)g(10)f(+)h(41\(mo)r(d)27 b(49\))1707 3487 y(=)22
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3864 y(mind,)28 b(w)n(e)f(can)g(pro)n(v)n(e)f(it)i(\(mo)r(d)g
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(e)456 4080 y(\(7\))1278 4024 y(1)p 1278 4061 V 1278
4137 a(1)1348 4080 y(+)1440 4024 y(1)p 1440 4061 V 1440
4137 a(2)1510 4080 y(+)1603 4024 y(1)p 1603 4061 V 1603
4137 a(3)1673 4080 y(+)18 b Fj(:)c(:)g(:)19 b Fn(+)2036
4024 y(1)p 1965 4061 185 4 v 1965 4137 a Fj(p)f Fi(\000)g
Fn(1)2182 4080 y(=)23 b(0)k(\(mo)r(d)h Fj(p)p Fn(\))456
4301 y(If)37 b(nothing)g(else,)i(it)e(will)g(giv)n(e)f(us)h(something)g
(to)g(do.)64 b(\(Besides,)39 b(if)f(w)n(e)e(can't)h(solv)n(e)f(this)456
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n(y)f(that)i(w)n(e)f(will)g(b)r(e)h(able)f(to)g(solv)n(e)f(the)i(\(mo)r
(d)g Fj(p)3154 4370 y Fl(2)3191 4401 y Fn(\))f(prob-)456
4500 y(lem.\))456 4718 y(It)f(turns)f(out)h(that)g(the)g(simpler)f
(problem)g(\(7\))h(is)g(m)n(uc)n(h)f(easier)g(to)g(w)n(ork)g(out.)36
b(F)-7 b(or)25 b(example,)456 4818 y(when)i Fj(p)h Fn(is)f(5,)h(w)n(e)f
(ha)n(v)n(e)1234 4973 y(1)p 1234 5010 42 4 v 1234 5086
a(1)1304 5029 y(+)1397 4973 y(1)p 1397 5010 V 1397 5086
a(2)1467 5029 y(+)1560 4973 y(1)p 1560 5010 V 1560 5086
a(3)1630 5029 y(+)1723 4973 y(1)p 1723 5010 V 1723 5086
a(4)1797 5029 y(=)c(1)18 b(+)g(3)g(+)g(2)g(+)g(4)27 b(\(mo)r(d)h(5\))
1797 5182 y(=)23 b(0)k(\(mo)r(d)h(5\))p eop end
%%Page: 27 41
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(e)768 586 y(1)p 768 623 42 4 v 768 699 a(1)838 642 y(+)931
586 y(1)p 931 623 V 931 699 a(2)1001 642 y(+)1094 586
y(1)p 1094 623 V 1094 699 a(3)1164 642 y(+)1257 586 y(1)p
1257 623 V 1257 699 a(4)1327 642 y(+)1419 586 y(1)p 1419
623 V 1419 699 a(5)1489 642 y(+)1582 586 y(1)p 1582 623
V 1582 699 a(6)1662 642 y(\(mo)r(d)g(7\))22 b(=)h(1)18
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794 y(=)23 b(1)18 b(+)g(2)g(+)g(3)g(+)g(4)g(+)g(5)g(+)g(6)27
b(\(mo)r(d)h(7\))1977 924 y(=)23 b(0)k(\(mo)r(d)h(7\))p
Fj(:)456 1086 y Fn(No)n(w)j(w)n(e)h(ha)n(v)n(e)f(a)h(pattern)h
(emerging:)45 b(the)32 b(recipro)r(cals)f(1)p Fj(=)p
Fn(1)p Fj(;)14 b Fn(1)p Fj(=)p Fn(2)p Fj(;)g(:)g(:)g(:)c(;)k
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Fj(p)p Fn(\))456 1186 y(seem)g(to)h(co)n(v)n(er)e(all)h(the)h(residues)
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Fj(p)19 b Fi(\000)g Fn(1\))29 b(\(mo)r(d)f Fj(p)p Fn(\))h(exactly)f
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1385 y(form)g(1)15 b(+)g(2)g(+)f(3)h(+)g(4)g(+)f(5)h(+)g(6,)26
b(whic)n(h)g(is)g(0.)36 b(T)-7 b(o)25 b(c)n(hec)n(k)g(a)h(lengthier)f
(example,)h(mo)r(d)h(11)e(yields)725 1521 y(1)p 725 1558
V 725 1634 a(1)795 1577 y(+)888 1521 y(1)p 888 1558 V
888 1634 a(2)958 1577 y(+)18 b Fj(:)c(:)g(:)k Fn(+)1270
1521 y(1)p 1249 1558 84 4 v 1249 1634 a(11)1365 1577
y(=)23 b(1)18 b(+)g(6)g(+)g(4)g(+)g(3)g(+)g(9)g(+)g(2)g(+)g(8)g(+)g(7)g
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b(1)18 b(+)g(2)g(+)g(3)g(+)g(4)g(+)g(5)g(+)g(6)g(+)g(7)g(+)g(8)g(+)g(9)
g(+)g(10)26 b(\(mo)r(d)j(11\))1365 1854 y(=)23 b(0)p
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2211 y(Instead)18 b(of)h(\015oundering)f(around)g(trying)g(to)h(\014t)g
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2311 y(it)33 b(can)f(b)r(e)i(done)e(if)h(y)n(ou)f(push)h(hard)g
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a(1)3222 2411 y Fn(+)3317 2378 y Fl(1)p 3317 2392 V 3317
2439 a(2)3380 2411 y Fn(+)466 2485 y Fl(1)p 466 2499
V 466 2546 a(3)525 2518 y Fn(+)16 b Fj(:)e(:)g(:)i Fn(+)853
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h Fj(p)3375 2488 y Fl(2)3412 2518 y Fn(\))456 2617 y(case.)456
2835 y(No)n(w)h(it)i(is)e(time)i(to)f(use)g(exp)r(erience)f(with)i
(these)f(sorts)e(of)i(problems.)44 b(F)-7 b(or)29 b(example,)h(if)h(w)n
(e)456 2935 y(are)24 b(fresh)i(from)f(solving)f(Problem)h(2.6,)g(w)n(e)
g(kno)n(w)g(that)h(symmetry)-7 b(,)26 b(or)e(an)n(ti-symmetry)h(can)456
3034 y(b)r(e)32 b(exploited,)i(esp)r(ecially)d(in)i(mo)r(dular)f
(arithmetic.)50 b(In)33 b(the)g(problem)e(of)i(pro)n(ving)e(\(7\))i(w)n
(e)456 3134 y(can)25 b(mak)n(e)g(the)h(sum)g(more)f(an)n(ti-symmetric)f
(b)n(y)i(replacing)e Fj(p)15 b Fi(\000)f Fn(1)25 b(with)i
Fi(\000)p Fn(1,)e Fj(p)14 b Fi(\000)h Fn(2)25 b(with)h
Fi(\000)p Fn(2,)456 3234 y(and)h(so)g(forth,)h(to)f(get)653
3369 y(1)p 653 3406 42 4 v 653 3482 a(1)723 3425 y(+)816
3369 y(1)p 816 3406 V 816 3482 a(2)886 3425 y(+)979 3369
y(1)p 979 3406 V 979 3482 a(3)1049 3425 y(+)18 b Fj(:)c(:)g(:)k
Fn(+)1411 3369 y(1)p 1340 3406 185 4 v 1340 3482 a Fj(p)g
Fi(\000)g Fn(1)1558 3425 y(=)1655 3369 y(1)p 1655 3406
42 4 v 1655 3482 a(1)1725 3425 y(+)1818 3369 y(1)p 1818
3406 V 1818 3482 a(2)1888 3425 y(+)1981 3369 y(1)p 1981
3406 V 1981 3482 a(3)2051 3425 y(+)g Fj(:)c(:)g(:)k Fn(+)2375
3369 y(1)p 2342 3406 107 4 v 2342 3482 a Fi(\000)p Fn(3)2477
3425 y(+)2602 3369 y(1)p 2570 3406 V 2570 3482 a Fi(\000)p
Fn(2)2704 3425 y(+)2830 3369 y(1)p 2797 3406 V 2797 3482
a Fi(\000)p Fn(1)2941 3425 y(\(mo)r(d)28 b Fj(p)p Fn(\))p
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(is)g(\\sort)g(of)6 b(".)43 b(When)30 b(w)n(e)f(solv)n(ed)g(the)h
(problem)f(\(mo)r(d)f Fj(p)p Fn(\),)j(w)n(e)e(paired)g(o\013)h(1)p
Fj(=)p Fn(1)456 4044 y(and)f(1)p Fj(=)p Fn(\()p Fj(p)19
b Fi(\000)h Fn(1\),)30 b(1)p Fj(=)p Fn(2)f(and)h(1)p
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b(When)31 b(w)n(e)e(try)h(the)g(same)f(pairing)g(in)456
4144 y(\(mo)r(d)f Fj(p)717 4113 y Fl(2)754 4144 y Fn(\),)g(what)f(w)n
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42 4 v 466 4397 a(1)535 4340 y(+)628 4284 y(1)p 628 4321
V 628 4397 a(2)698 4340 y(+)18 b Fj(:)c(:)g(:)19 b Fn(+)1061
4284 y(1)p 990 4321 185 4 v 990 4397 a Fj(p)f Fi(\000)g
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42 4 v 1337 4397 a(1)1407 4340 y(+)1572 4284 y(1)p 1500
4321 185 4 v 1500 4397 a Fj(p)18 b Fi(\000)h Fn(1)1695
4340 y(\))g(+)f(\()1871 4284 y(1)p 1871 4321 42 4 v 1871
4397 a(2)1941 4340 y(+)2106 4284 y(1)p 2034 4321 185
4 v 2034 4397 a Fj(p)g Fi(\000)g Fn(2)2229 4340 y(\))g(+)g
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333 4 v 2603 4397 a(\()p Fj(p)g Fi(\000)g Fn(1\))p Fj(=)p
Fn(2)2964 4340 y(+)3202 4284 y(1)p 3057 4321 V 3057 4397
a(\()p Fj(p)g Fn(+)g(1\))p Fj(=)p Fn(2)3399 4340 y(\))1207
4543 y(=)1480 4487 y Fj(p)p 1305 4524 393 4 v 1305 4600
a Fn(1)g Fi(\002)g Fn(\()p Fj(p)h Fi(\000)f Fn(1\))1726
4543 y(+)1994 4487 y Fj(p)p 1819 4524 V 1819 4600 a Fn(2)g
Fi(\002)g Fn(\()p Fj(p)g Fi(\000)g Fn(2\))2240 4543 y(+)g
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4 v 2531 4600 a Fn(\()p Fj(p)g Fi(\000)g Fn(1\))p Fj(=)p
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4774 y(=)23 b Fj(p)1351 4657 y Fe(\024)1580 4718 y Fn(1)p
1405 4755 393 4 v 1405 4831 a(1)17 b Fi(\002)h Fn(\()p
Fj(p)h Fi(\000)f Fn(1\))1825 4774 y(+)2094 4718 y(1)p
1918 4755 V 1918 4831 a(2)g Fi(\002)g Fn(\()p Fj(p)h
Fi(\000)f Fn(2\))2339 4774 y(+)g Fj(:)c(:)g(:)k Fn(+)2993
4718 y(1)p 2630 4755 767 4 v 2630 4831 a(\()p Fj(p)h
Fi(\000)f Fn(1\))p Fj(=)p Fn(2)f Fi(\002)h Fn(\()p Fj(p)h
Fn(+)f(1\))p Fj(=)p Fn(2)3406 4657 y Fe(\025)1295 4958
y Fn(\(mo)r(d)28 b Fj(p)1556 4924 y Fl(2)1593 4958 y
Fn(\))p Fj(:)456 5116 y Fn(No)n(w)k(this,)i(at)f(\014rst,)h(lo)r(oks)d
(lik)n(e)i(a)f(complication)g(rather)g(than)g(a)h(simpli\014cation.)52
b(But)33 b(w)n(e)456 5216 y(ha)n(v)n(e)d(gained)g(a)h(v)n(ery)f(imp)r
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48 b(No)n(w,)32 b(instead)f(of)p eop end
%%Page: 28 42
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(that)1476 559 y(\(expression)o(\))d(=)g(0)k(\(mo)r(d)h
Fj(p)2354 525 y Fl(2)2391 559 y Fn(\))456 685 y(w)n(e)f(no)n(w)g(ha)n
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829 y(\()p Fj(p)18 b Fi(\002)g Fn(expression)o(\))23
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Fn(\))456 972 y(whic)n(h)f(is)h(equiv)-5 b(alen)n(t)27
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1677 y(And)32 b(it)f(is)h(quic)n(kly)e(seen)h(that)h(the)g(apparen)n(t)
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1776 y(illusionary)-7 b(,)22 b(as)h(the)g(\(mo)r(d)28
b Fj(p)p Fn(\))23 b(can)g(get)f(rid)h(of)g(a)g(lot)g(more)f(terms)g
(than)i(\(mo)r(d)k Fj(p)2985 1746 y Fl(2)3022 1776 y
Fn(\))23 b(can.)35 b(No)n(w,)456 1876 y(w)n(e)27 b(only)g(ha)n(v)n(e)f
(to)i(sho)n(w)f(that)881 1998 y(1)p 706 2035 393 4 v
706 2111 a(1)18 b Fi(\002)g Fn(\()p Fj(p)g Fi(\000)h
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2111 a(2)f Fi(\002)g Fn(\()p Fj(p)g Fi(\000)g Fn(2\))1640
2054 y(+)g Fj(:)c(:)g(:)19 b Fn(+)2294 1998 y(1)p 1932
2035 767 4 v 1932 2111 a(\()p Fj(p)f Fi(\000)g Fn(1\))p
Fj(=)p Fn(2)g Fi(\002)g Fn(\()p Fj(p)g Fn(+)g(1\))p Fj(=)p
Fn(2)2731 2054 y(=)23 b(0)k(\(mo)r(d)h Fj(p)p Fn(\))p
Fj(:)456 2246 y Fn(But)i Fj(p)21 b Fi(\000)e Fn(1)31
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b(alen)n(t)30 b(to)h Fi(\000)p Fn(2)e(\(mo)r(d)f Fj(p)p
Fn(\),)k(and)e(so)456 2346 y(forth,)d(so)g(the)h(equation)f(reduces)g
(to)1140 2468 y(1)p 1089 2505 144 4 v 1089 2581 a Fi(\000)p
Fn(1)1196 2557 y Fl(2)1260 2524 y Fn(+)1404 2468 y(1)p
1353 2505 V 1353 2581 a Fi(\000)p Fn(2)1460 2557 y Fl(2)1525
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1816 2505 499 4 v 1816 2581 a Fi(\000)p Fn(\(\()p Fj(p)h
Fi(\000)f Fn(1\))p Fj(=)p Fn(2\))2279 2557 y Fl(2)2348
2524 y Fn(=)23 b(0)k(\(mo)r(d)h Fj(p)p Fn(\))p Fj(;)456
2712 y Fn(or)e(equiv)-5 b(alen)n(tly)1104 2834 y(1)p
1085 2871 79 4 v 1085 2947 a(1)1127 2923 y Fl(2)1193
2890 y Fn(+)1304 2834 y(1)p 1286 2871 V 1286 2947 a(2)1328
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Fn(\))p Fj(:)p eop end
%%Page: 29 43
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end
%%Page: 30 44
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%%Trailer
userdict /end-hook known{end-hook}if
%%EOF