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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 26 "  Elementary Number Theory" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 312 "Save you
r current worksheet as number2.mws.  Remember to save frequently in ca
se of system shut downs in the middle of a work session! In the text m
ode (click on the cap T button) place your name(s) and date at the top
 of your worksheet.  At the end of this session you will print out and
 turn in your worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 255 "We continue by experimenting with some more very \+
basic calculations.  Again the intent is to learn some basic MAPLE syn
tax, while observing more neat features of a Computer Algebra System (
CAS).  We will also review some basic concepts from number theory." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "Preform \+
the following calculations with MAPLE.  You must reenter the calculati
on mode (click on the [> button). As you do these exercises enter note
s and comments onto your worksheet in text mode." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "gcd(6,8);" "6#
-%$gcdG6$\"\"'\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 35 "1.  What is gcd a
n abreviation for?" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "lcm(6,8)
;" "6#-%$lcmG6$\"\"'\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 35 "2.  What is
 lcm an abreviation for?" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "gc
d(6,8)*lcm(6,8);" "6#*&-%$gcdG6$\"\"'\"\")\"\"\"-%$lcmG6$\"\"'\"\")F)
" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "gcd(140,958);" "6#-%$gcdG
6$\"$S\"\"$e*" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "lcm(140,958)
;" "6#-%$lcmG6$\"$S\"\"$e*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
5 "%*%%;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "140*958;" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 " 3.   Wh
at general formula is suggested by these calculations?    Test the for
mula with several more pairs of integers.  " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "     Continue the exercises." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 
1 "gcd(101+45,36^2);" "6#-%$gcdG6$,&\"$,\"\"\"\"\"#XF(*$\"#O\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "lcm(5*(1+4^3),2^5);" "6#-%$lcm
G6$*&\"\"&\"\"\",&\"\"\"F(*$\"\"%\"\"$F(F(*$\"\"#\"\"&" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "36/gcd(36,16);" "6#*&\"#O\"\"\"-%$gcd
G6$\"#O\"#;!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "lcm(38,10
0)/gcd(38,100);" "6#*&-%$lcmG6$\"#Q\"$+\"\"\"\"-%$gcdG6$\"#Q\"$+\"!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "lcm(1524,5587);" "6#-%$
lcmG6$\"%C:\"%(e&" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "lcm(1524
,5588);" "6#-%$lcmG6$\"%C:\"%)e&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 71 "      Notice that bigger integers don't n
ecessarily yield bigger lcm's." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "gcd(14^6,15^20);" "6#-%$gcdG6$
*$\"#9\"\"'*$\"#:\"#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 95 "     Two integers are \"relatively prime\" if the bi
ggest integer that divides both of them is 1." }}{PARA 0 "" 0 "" 
{TEXT -1 80 "4.  List all the pairs of integers from 100 to 110 that a
re relatively prime.   " }{TEXT 256 109 "Here you might want to ask fo
r the MAPLE Programming Supplement to learn how to do \"for\" and \"if
\" programs.  " }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }}{PARA 0 "" 0 "
" {TEXT -1 74 "     An integer is called \"prime\" if its only divisor
s are 1 and itself.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 182 "     The MAPLE command \"isprime\" determines whether
 or not a given integer is prime.  Use it to see if the following numb
ers are prime:  29,  35,  1537,  your ss#,   10!+1,  329891. " }}
{PARA 0 "" 0 "" {TEXT -1 46 "5.  List the prime numbers between 1 and \+
100. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 
"     The FUNDAMENTAL THEOREM OF ARITHEMATIC states that every integer
 can be factored in a unique way into a product of powers of primes.  \+
" }}{PARA 0 "" 0 "" {TEXT -1 120 "6.  Use paper and pencil to factor t
he following into their prime decompositions:  10! ,  340704.  Include
 your answers." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 59 "     The MAPLE command \"ifactor\" does this automaticall
y.  " }}{PARA 0 "" 0 "" {TEXT -1 101 "7.  Use it to verify your result
s from above.  Now factor the following numbers:  10!+1 ,  your ss#. \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Execu
te the following command sequence:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "2744280=ifactor(2744280);\n267300=ifactor(267300);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "2744280*267300=ifactor(274
4280*267300);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "gcd(274428
0,267300)=ifactor(gcd(2744280,267300));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "lcm(2744280,267300)=ifactor(lcm(2744280,267300));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "lcm(2744280,267300)*gcd(274
4280,267300)=ifactor(lcm(2744280,267300)*gcd(2744280,267300));" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 251 " 8.  Usi
ng the FUNDAMENTAL THEOREM OF ARITHEMATIC, provide a written argument \+
to verify the formula:  gcd(m,n)lcm(m,n) = mn.  Demonstrate your argum
ent with two sufficiently interesting integers (different from above) \+
and the ifactor command in MAPLE." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 95 "     For two integers m & n  with  n > or
 = m we say the \"remainder of n divided by m\" is r if " }}{PARA 0 "
" 0 "" {TEXT -1 58 "n = qm + r, for integers q >0 & r with 0 < or = r \+
<  m.   " }}{PARA 0 "" 0 "" {TEXT -1 157 "9.  With paper and pencil co
mpute the following remainders and include your answers:  24 divided b
y 5,  156 divided by 7,  32 divided by 2,  57 divided by 4." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "      In MAPLE
 this remainder is denoted by \"n mod m.\" Verify the above answers wi
th MAPLE and compute the following remainders:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "13200 mod 134;
" "6#-%$modG6$\"&+K\"\"$M\"" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 
1 "(12^3+53) mod 26;" "6#-%$modG6$,&*$\"#7\"\"$\"\"\"\"#`F*\"#E" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "400*23 mod 19;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "1321 mod (5*7);" }}}{EXCHG {PARA 0 
"> " 0 "" {XPPEDIT 19 1 "45^16 mod 4;" "6#-%$modG6$*$\"#X\"#;\"\"%" }}
}{PARA 0 "" 0 "" {TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT -1 5 "    \+
 " }}{PARA 0 "" 0 "" {TEXT -1 93 "10.   Explain why an integer n is \"
even\" exactly when n mod 2 = 0, and is \"odd\" exactly when " }}
{PARA 0 "" 0 "" {TEXT -1 12 "n mod 2 = 1." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "11.   Explain why every product o
f two odd integers must be an odd integer and every product of an even
 integer with another integer must be even. " }}{PARA 0 "" 0 "" {TEXT 
-1 102 "     Do the following calculations in your head and record the
 answers.  Check the results with MAPLE." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "1342 mod 2;" "6#-%$modG
6$\"%U8\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "(10!+1) mod 2
;" "6#-%$modG6$,&-%*factorialG6#\"#5\"\"\"\"\"\"F+\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "18^11 mod 2;" "6#-%$modG6$*$\"#=\"#6
\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "`mod`(23^10,2);" "6#
-%$modG6$*$\"#B\"#5\"\"#" }}}}{MARK "72 0" 3 }{VIEWOPTS 1 1 0 1 1 
1803 }