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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 52 "Abstract Algebra with Map
le\nChapter 1: Preliminaries" }}{PARA 19 "" 0 "" {TEXT -1 3 "by " }
{URLLINK 17 "Alec Mihailovs" 4 "http://webpages.shepherd.edu/amihailo
" "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 7 "Preface" }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 40 "This manual is intended to be used with " }
{URLLINK 17 "Contemporary Abstract Algebra" 4 "http://college.hmco.com
/mathematics/gallian/abstract_algebra/5e/students/" "" }{TEXT -1 5 ", \+
by " }{URLLINK 17 "J. A. Gallian" 4 "http://www.d.umn.edu/~jgallian/" 
"" }{TEXT -1 55 ", 5th ed., Houghton Mifflin (2002). It was inspired b
y " }{URLLINK 17 "Abstract Algebra with GAP" 4 "http://college.hmco.co
m/mathematics/gallian/abstract_algebra/5e/students/gap.html" "" }
{TEXT -1 4 " by " }{URLLINK 17 "J. G. Rainbolt" 4 "http://euler.slu.ed
u/Dept/Faculty/rainbolt/rainbolt.html" "" }{TEXT -1 5 " and " }
{URLLINK 17 "J. A. Gallian" 4 "http://www.d.umn.edu/~jgallian/" "" }
{TEXT -1 254 ", Houghton Mifflin (2002). The latter manual is availabl
e for free downloading from the book's web site and from the authors` \+
websites. I highly recommend, in addition to this Maple manual, solvin
g exercises from both the textbook and the GAP manual.    " }}}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 16 "1: Preliminaries" }}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 22 "Properties of Integers" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 40 "It is easy to factor integers in Maple: " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ifactor(2^57-1);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#**-%!G6#\"\"(\"\"\"-F%6#\"(ZG@\"F(-F%6#\"'(GC&F(-F%6#
\"&xB$F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "The greatest common d
ivisor and the least common multiple can be evaluated in Maple as foll
ows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "igcd(715,1001);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$V\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "ilcm(843,216,51);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"(K=.\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The next example show
s how to find integer solutions of equations: " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "isolve(7*x+15*y=1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$/%\"xG,&!\"#\"\"\"*&\"#:F(%$_Z1GF(!\"\"/%\"yG,&F(F(*&
\"\"(F(F+F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "To find a partic
ular solution, one can replace the unknown " }{XPPEDIT 18 0 "_Z1;" "6#
%$_Z1G" }{TEXT -1 41 " by any integer value, for example, by 2:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "subs(_Z1=2,%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#<$/%\"xG!#K/%\"yG\"#:" }}}}{SECT 0 {PARA 4 "" 0 
"" {TEXT -1 18 "Modular Arithmetic" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
38 " Maple can do modular arithmetic, too:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "345 mod 7;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "It can be used for checking \+
the validity of money order numbers, UPS pickup record numbers, ISBN n
umbers etc. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "isMoneyOrde
r:= n -> n<10^11 and trunc(n/10) mod 9 = n mod 10:" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 187 "This function first checks if the number contain
s not more than 11 digits and then if the number formed by the first 1
0 digits is congruent to the last digit, i.e. check digit, modulo 9. \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "isMoneyOrder(3953988164
2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "isMoneyOrder(39559881642);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "The
 similar construction works for air ticket numbers and UPS pickup reco
rds numbers, just by replacing modulo 9 to modulo 7:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 54 "isUPS:= n -> n<10^10 and trunc(n/10) mod \+
7 = n mod 10:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "isUPS(7681
139992);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "isUPS(1213731473673);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
60 "isAirTicket:= n -> n<10^15 and trunc(n/10) mod 7 = n mod 10:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "isAirTicket(1213731473673);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 102 "For the UPC code, one needs to use a dot product, so w
e have to load the Linear Algebra package first:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 102 "isUPC:= n -> evalb(Vector[row](12,convert(n,bas
e,10)) . Vector([1,3,1,3,1,3,1,3,1,3,1,3]) mod 10 = 0):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "isUPC(021000658978);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "isUPC(012000658978);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "A similar construction works \+
for bank checks:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "isBank
Check:= n -> evalb(Vector[row](9,convert(n,base,10)) . Vector([9,3,7,9
,3,7,9,3,7]) mod 10 = 0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "isBankCheck(13);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "For ISBN numbers we need a slight
ly more sophisticated method, because inputs can include the letter X:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "isISBN:= proc(str) loc
al s;\nwith(LinearAlgebra);\nif type(str,integer) then s:=[str]; else
\ns:=sscanf(str,\"%d%[Xx]\") end if;\nevalb(`if`(nops(s)=1,Vector[row]
(10,convert(s[1],base,10)),\n`if`(nops(s)=2 and not s[2]=\"\", Vector[
row](10, [10,op(convert(s[1],base,10))]),Vector[row]([1,0$9])))  . Vec
tor([$ 1..10]) mod 11 = 0) end: " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "isISBN(0618122141);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "isISBN(\"61812
2141\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 13 "isISBN(\"6x\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 237 "As you
 can see, ISBN numbers without an X can be entered either as integers,
 or as strings, inside quotes. ISBN numbers with an X at the end must \+
be entered inside quotes, because it is not one of the data formats th
at Maple recognizes. " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 "Mathem
atical Induction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Maple can eval
uate many sums:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "sum(i,i=
1..n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$),&%\"nG\"\"\"F(F(\"\"#F
(#F(F)*&#F(F)F(F'F(!\"\"#F(F)F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"nG\"\"
#\"\"\"#F(F'*&F)F(F&F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"simplify(sum(i^10,i=1..n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$)%
\"nG\"\"$\"\"\"#!\"\"\"\"#*$)F&\"\"&F(F(*$)F&\"\"(F(F**&#F.\"\"'F()F&
\"\"*F(F(*&#F(F+F()F&\"#5F(F(*&#F(\"#6F()F&F=F(F(*&#F.\"#mF(F&F(F(" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sort(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,0*$)%\"nG\"#6\"\"\"#F(F'*&#F(\"\"#F()F&\"#5F(F(*&#\"
\"&\"\"'F()F&\"\"*F(F(*$)F&\"\"(F(!\"\"*$)F&F1F(F(*&#F(F,F(*$)F&\"\"$F
(F(F8*&#F1\"#mF(F&F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The ans
wers can be proven by mathematical induction as follows: " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f:=n->1/11*n^11+1/2*n^10+5/6*n^9-n^
7+n^5-1/2*n^3+5/66*n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f(
1)=1 and simplify(f(n+1))=simplify(f(n)+(n+1)^10);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Also
, Maple can find formulas for many polynomial sequences, for example, \+
sums of squares, 1, 5, 14, 30, 55, ..., using interpolating polynomial
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "interp([$1..5],[1,5,14
,30,55],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"$\"\"\"#F
(F'*&#F(\"\"#F()F&F,F(F(*&#F(\"\"'F(F&F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 64 "Since formula is known, Maple can easily continue the seq
uence: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "f:=x->1/3*x^3+1/
2*x^2+1/6*x: \nfor N from 6 to 10 do f(N)  od; " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$S\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$/#" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"$&G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$&Q" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Exercises" }}{EXCHG {PARA 
14 "" 0 "" {TEXT -1 0 "" }{TEXT 256 1 "1" }{TEXT -1 9 ". Factor " }
{XPPEDIT 18 0 "3^100+1;" "6#,&*$\"\"$\"$+\"\"\"\"F'F'" }{TEXT -1 1 ".
" }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 257 1 "2" }{TEXT -1 68 ". Fi
nd the greatest common divisor and the least common multiple of " }
{XPPEDIT 18 0 "670592745;" "6#\"*XFfq'" }{TEXT -1 2 ", " }{XPPEDIT 18 
0 "83810205;" "6#\")0-\"Q)" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "113351790
;" "6#\"*!z^L6" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "695529645;" "6#\"
*X'Hbp" }{TEXT -1 2 ". " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 258 
1 "3" }{TEXT -1 41 ". Find integer solutions of the equation " }
{XPPEDIT 18 0 "42*x-47*y = 1;" "6#/,&*&\"#U\"\"\"%\"xGF'F'*&\"#ZF'%\"y
GF'!\"\"F'" }{TEXT -1 1 "." }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 
259 1 "4" }{TEXT -1 68 ". Find the last digit of the ISBN number start
ing from 1-894511-01. " }}{PARA 14 "" 0 "" {TEXT -1 0 "" }{TEXT 260 1 
"5" }{TEXT -1 67 ". Find the formula for the sum of 9th powers of inte
gers from 1 to " }{TEXT 261 1 "n" }{TEXT -1 2 ". " }}{PARA 14 "" 0 "" 
{TEXT -1 0 "" }{TEXT 262 1 "6" }{TEXT -1 129 ". Find the formula for t
he elements of the sequence 3, 17, 81, 255, 623, 1293, ... and find th
e next 4 elements of the sequence. " }}}}}}{MARK "0 0 0" 0 }{VIEWOPTS 
1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }