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{SECT 0 {EXCHG {PARA 211 "" 0 "" {TEXT -1 31 "                        \+
       " }{TEXT -1 1 " " }{TEXT 206 28 "USING MAPLE IN NUMBER THEORY" 
}}{PARA 212 "" 0 "" }{PARA 212 "" 0 "" {TEXT -1 198 "Maple has several
 functions that are useful in solving problems involving integers.  Th
e first batch is loaded with the Maple kernel (i.e., they can be used \+
immediately without further preparation.)" }}{PARA 212 "" 0 "" }{PARA 
212 "" 0 "" {TEXT -1 43 "1.  To find the (n,k) binomial coefficient." 
}}{PARA 212 "" 0 "" {TEXT -1 18 "     binomial(n,k)" }}{PARA 213 "> " 
0 "" {MPLTEXT 1 0 15 "binomial(10,3);" }}{PARA 214 "" 1 "" {XPPMATH 
20 "6#\"$?\"" }}}{EXCHG {PARA 212 "" 0 "" {TEXT -1 32 "2.  To generate
 a random number." }}{PARA 212 "" 0 "" {TEXT -1 58 "      rand() gener
ates a 12 digit positive random integer." }}{PARA 213 "> " 0 "" 
{MPLTEXT 1 0 9 "rand();  " }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"-Z;`'*
z#*" }}}{EXCHG {PARA 212 "" 0 "" {TEXT -1 126 "    rand(m..n) creates \+
a program to generate a random integer between m and n.  rand()     th
en elicits numbers in this range." }}{PARA 213 "> " 0 "" {MPLTEXT 1 0 
17 "die :=rand(1..6):" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 0 27 "
die(); die(); die(); die();" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"\"
" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}
{EXCHG {PARA 212 "" 0 "" {TEXT -1 57 "Change the seed to change the se
quence of random numbers." }}{PARA 213 "> " 0 "" {MPLTEXT 1 0 12 "_see
d :=100:" }}}{EXCHG {PARA 212 "" 0 "" {TEXT -1 85 "randomize() sets th
e seed to depend on the time current when the command was entered." }}
}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 0 12 "randomize();" }}{PARA 
214 "" 1 "" {XPPMATH 20 "6#\"+kv`X5" }}}{EXCHG {PARA 212 "" 0 "" 
{TEXT -1 54 "3.  To employ the division algorithm to divide b by a." }
}{PARA 212 "" 0 "" {TEXT -1 71 "     iquo(b,a,r) returns the quotient \+
q; enter r to find the remainder." }}{PARA 213 "> " 0 "" {MPLTEXT 1 0 
15 "iquo(19,4,'r');" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}
{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 0 2 "r;" }}{PARA 214 "" 1 "" 
{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 212 "" 0 "" {TEXT -1 65 "4.  To \+
employ the Euclidean algorithm to find the gcd of a and b." }}{PARA 
212 "" 0 "" {TEXT -1 14 "     igcd(a,b)" }}{PARA 213 "> " 0 "" 
{MPLTEXT 1 0 13 "igcd(326,78);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"
\"#" }}}{EXCHG {PARA 212 "" 0 "" {TEXT -1 109 "Another command is nece
ssary in order to express the gcd as a linear combination of a and b: \+
(a,b) = sa + tb." }}{PARA 213 "> " 0 "" {MPLTEXT 1 0 19 "igcdex(326,78
,s,t);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 213 
"> " 0 "" {MPLTEXT 1 0 4 "s;t;" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#!#6
" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"#Y" }}}{EXCHG {PARA 212 "" 0 ""
 {TEXT -1 31 "5.  To find the lcm of a and b." }}{PARA 212 "" 0 "" 
{TEXT -1 14 "     ilcm(a,b)" }}{PARA 213 "> " 0 "" {MPLTEXT 1 0 13 "il
cm(326,78);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"&9F\"" }}}{EXCHG 
{PARA 212 "" 0 "" {TEXT -1 39 "6.  To factor the number n into primes.
" }}{PARA 212 "" 0 "" {TEXT -1 17 "      ifactors(n)" }}{PARA 213 "> "
 0 "" {MPLTEXT 1 0 13 "ifactor(100);" }}{PARA 214 "" 1 "" {XPPMATH 20 
"6#*&)-%!G6#\"\"#F(\"\"\")-F&6#\"\"&F(F)" }}}{EXCHG {PARA 213 "> " 0 "
" {MPLTEXT 1 0 20 "ifactor(1234567890);" }}{PARA 214 "" 1 "" {XPPMATH 
20 "6#*,-%!G6#\"\"#\"\"\")-F%6#\"\"$F'F(-F%6#\"\"&F(-F%6#\"%.QF(-F%6#
\"%2OF(" }}}{EXCHG {PARA 212 "" 0 "" {TEXT -1 33 "7.  To find the ith \+
prime number." }}{PARA 212 "" 0 "" {TEXT -1 17 "      ithprime(i)" }}
{PARA 213 "> " 0 "" {MPLTEXT 1 0 12 "ithprime(4);" }}{PARA 214 "" 1 ""
 {XPPMATH 20 "6#\"\"(" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 0 15 "
ithprime(1000);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#\"%>z" }}}{EXCHG 
{PARA 212 "" 0 "" {TEXT -1 69 "THE FOLLOWING COMMANDS REQUIRE THAT THE
 NUMTHEORY PACKAGE BE LOADED. " }}{PARA 213 "> " 0 "" {MPLTEXT 1 0 16 
"with(numtheory):" }}{PARA 215 "" 1 "" {TEXT 207 68 "Warning, the prot
ected name order has been redefined and unprotected" }{TEXT 207 1 "\n"
 }}}{EXCHG {PARA 212 "" 0 "" {TEXT -1 50 "8.  To find the set of prime
 factors of integer n." }}{PARA 212 "" 0 "" {TEXT -1 17 "     factorse
t(n)" }}{PARA 213 "> " 0 "" {MPLTEXT 1 0 15 "factorset(100);" }}{PARA 
214 "" 1 "" {XPPMATH 20 "6#<$\"\"#\"\"&" }}}{EXCHG {PARA 212 "" 0 "" }
{PARA 213 "> " 0 "" {MPLTEXT 1 0 22 "factorset(1234567890);" }}{PARA 
214 "" 1 "" {XPPMATH 20 "6#<'\"\"#\"\"$\"\"&\"%2O\"%.Q" }}}{EXCHG 
{PARA 212 "" 0 "" {TEXT -1 61 "9.  To obtain the complete list of fact
ors of integer n.     " }}{PARA 213 "> " 0 "" {MPLTEXT 1 0 14 "divisor
s(100);" }}{PARA 214 "" 1 "" {XPPMATH 20 "6#<+\"\"\"\"\"#\"\"%\"\"&\"#
5\"#?\"#D\"#]\"$+\"" }}}{EXCHG {PARA 213 "> " 0 "" {MPLTEXT 1 0 21 "di
visors(1234567890);" }}{PARA 216 "" 1 "" {XPPMATH 20 "6#<R\"\"\"\"\"#
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