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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 12 "Assignment 2" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 268 5 "Note:" }{TEXT 287 1 " " }{TEXT 288 106 "While w
riting any procedure, you'd better declare local variables in the body
 of the procedure and end the" }{TEXT -1 0 "" }{TEXT 282 35 " procedur
e definition with a colon." }{TEXT 260 2 "\n\n" }}{PARA 0 "" 0 "" 
{TEXT 290 10 "Problem 1." }{TEXT -1 61 "  \n(a) Write a program (not n
ecessarily a procedure) using a " }{TEXT 276 7 "do loop" }{TEXT -1 20 
" to compute the sum " }{XPPEDIT 18 0 "sum((2/3)^n,n = 2 .. 100);" "6#
-%$sumG6$)*&\"\"#\"\"\"\"\"$!\"\"%\"nG/F,;F(\"$+\"" }{TEXT -1 9 ".  Ap
ply " }{TEXT 261 5 "evalf" }{TEXT -1 54 " to obtain a floating point a
pproximation to the sum.\n" }{MPLTEXT 1 0 0 "" }{TEXT -1 70 "\n(b) Wri
te a program (not necessarily a procedure) that will find the " }
{TEXT 262 3 "set" }{TEXT -1 107 " S of all positive integers less than
 1000 that have 11 as a factor.  [n has 11 as a factor if and only if \+
" }{XPPEDIT 18 0 "`mod`(n,11) = 0;" "6#/-%$modG6$%\"nG\"#6\"\"!" }
{TEXT -1 2 ".]" }}{PARA 0 "" 0 "" {TEXT 258 1 "\n" }}{PARA 0 "" 0 "" 
{TEXT 289 11 "Problem 2. " }{TEXT -1 20 "\n\nWrite a procedure " }
{TEXT 269 7 "ISPRIME" }{TEXT -1 57 " whose input is a positive integer
 n and whose output is " }{TEXT 274 4 "true" }{TEXT -1 19 " if n is pr
ime and " }{TEXT 275 5 "false" }{TEXT -1 58 " otherwise.  Test the pro
cedure on the following numbers: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 45 "1, 2, 3, 4, 11, 111, 531, 7919, 15689, 26
2681" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "T
he answers should be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {XPPEDIT 18 0 "false,true,true,false,true,false,false,true,false,tru
e;" "6,%&falseG%%trueGF$F#F$F#F#F$F#F$" }{TEXT 264 1 "." }}{PARA 0 "" 
0 "" {TEXT 286 6 "\nHint:" }{TEXT -1 36 " Here's a start for the progr
am:\n\n> " }{MPLTEXT 1 0 111 " ISPRIME:=proc(n)\n   local d;\n   if n \+
< 2 then return false; end if;\n   if n = 2 then return true; end if;
\n   \n" }{TEXT -1 13 "Note that if " }{XPPEDIT 18 0 "n;" "6#%\"nG" }
{TEXT -1 33 " < 2 then the third  line causes " }{TEXT 270 10 "ISPRIME
(n)" }{TEXT -1 21 " to return false. If " }{XPPEDIT 18 0 "n;" "6#%\"nG
" }{TEXT -1 33 " = 2 then the fourth line causes " }{TEXT 271 7 "ISPRI
ME" }{TEXT -1 246 " to return true. If neither is true then n is at le
ast 3.  So now you only need to handle in the rest of the program the \+
case that n is at least 3. This can be done by using a loop to check w
hether or not n is divisible by an integer d from 2 to " }{XPPEDIT 18 
0 "n-1;" "6#,&%\"nG\"\"\"F%!\"\"" }{TEXT -1 175 ". If you find such a \+
value of d you want to return false. If the loop is completed without \+
finding any such d then you should return true. You may be able to imp
rove on this.\n" }}{PARA 0 "" 0 "" {TEXT 263 6 "Remark" }{TEXT -1 48 "
. Maple actually has a built-in procedure named " }{TEXT 267 7 "isprim
e" }{TEXT -1 47 " which performs the same task as the procedure " }
{TEXT 272 7 "ISPRIME" }{TEXT -1 16 " in Problem 2.  " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Maple's procedure " }
{TEXT 279 7 "isprime" }{TEXT -1 13 " is really a " }{TEXT 281 31 "prob
abilistic primality tester." }{TEXT -1 5 " See " }{TEXT 280 8 "?isprim
e" }{TEXT -1 28 ".  To see how the procedure " }{TEXT 278 7 "isprime" 
}{TEXT -1 36 " is constructed execute the command " }{TEXT 277 19 "sho
wstat(isprime); " }{TEXT -1 145 "However, it is not easy to see what t
he program is doing. So I don't recommend spending a lot of time on tr
ying to figure it out unless you have " }{TEXT 283 4 "lots" }{TEXT -1 
48 " of experience in programming and number theory." }}{PARA 0 "" 0 "
" {TEXT -1 1 "\n" }{TEXT 259 10 "Problem 3." }{TEXT -1 27 "  \n\n(a) W
rite a procedure, " }{TEXT 256 8 "primesum" }{TEXT -1 36 ", whose inpu
t is a positive integer " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 43 "
 and whose output is the sum of all primes " }{XPPEDIT 18 0 "p;" "6#%
\"pG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "p <= n;" "6#1%\"pG%\"
nG" }{TEXT -1 37 ". You may use either your procedure  " }{TEXT 273 7 
"ISPRIME" }{TEXT -1 42 " from problem 2 or the built-in procedure " }
{TEXT 257 9 "isprime. " }{TEXT -1 23 "Test the procedure for " }
{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 22 " = 2, 4, 10, and 1000." }}
{PARA 0 "" 0 "" {TEXT -1 25 "\n (b) Write a procedure, " }{TEXT 265 8 
"primeset" }{TEXT -1 37 ", whose input is  a positive integer " }
{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 25 " and whose output is the " 
}{TEXT 291 3 "set" }{TEXT -1 15 " of all primes " }{XPPEDIT 18 0 "p;" 
"6#%\"pG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "p <= n;" "6#1%\"p
G%\"nG" }{TEXT -1 87 ". You may use either your procedure  ISPRIME fro
m problem 2  or the built-in procedure " }{TEXT 266 9 "isprime. " }
{TEXT -1 23 "Test the procedure for " }{XPPEDIT 18 0 "n;" "6#%\"nG" }
{TEXT -1 22 " = 2, 4, 10, and 1000." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 292 9 "Problem 4" }}{PARA 0 "" 
0 "" {TEXT -1 93 "You should remember from your algebra classes that q
uadratic equations have the general form " }{XPPEDIT 18 0 "a*x^2+b*x+c
 = 0;" "6#/,(*&%\"aG\"\"\"*$)%\"xG\"\"#F'F'F'*&%\"bGF'F*F'F'%\"cGF'\"
\"!" }{TEXT -1 55 ", and the solutions are given by the quadratic form
ula:" }}{PARA 0 "" 0 "" {TEXT -1 20 "                    " }{XPPEDIT 
18 0 "x = (-b+sqrt(b^2-4*a*c))/(2*a);" "6#/%\"xG*&,&%\"bG!\"\"-%%sqrtG
6#,&*$)F'\"\"#\"\"\"F0*(\"\"%F0%\"aGF0%\"cGF0F(F0F0*&F/F0%\"aGF0F(" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "x = (-b-sqrt(b^2-4*a*c))/(2*a);" "6
#/%\"xG*&,&%\"bG!\"\"-%%sqrtG6#,&*$)F'\"\"#\"\"\"F0*(\"\"%F0%\"aGF0%\"
cGF0F(F(F0*&F/F0F3F0F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Write a prcedure cal
led QSOLVER that receives the values of a,b,and c and outputs the two \+
solutions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
284 13 "Extra Credit." }{TEXT -1 26 " \nThree positive integers " }
{XPPEDIT 18 0 "x,y;" "6$%\"xG%\"yG" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "z;" "6#%\"zG" }{TEXT -1 28 " form a Pathagorean triple (" }
{XPPEDIT 18 0 "x,y,z;" "6%%\"xG%\"yG%\"zG" }{TEXT -1 5 ") if " }
{XPPEDIT 18 0 "x^2+y^2 = z^2;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%
\"zGF'" }{TEXT -1 66 ". For example, (3,4,5) is a Pathagorean triple. \+
Write a procedure " }{TEXT 285 20 "Pathagoreantripleset" }{TEXT -1 36 
", whose input is a positive integer " }{XPPEDIT 18 0 "n;" "6#%\"nG" }
{TEXT -1 57 " and whose output is the set of all Pathagorean triples (
" }{XPPEDIT 18 0 "x,y,z;" "6%%\"xG%\"yG%\"zG" }{TEXT -1 7 ") with " }
{XPPEDIT 18 0 "z <= n;" "6#1%\"zG%\"nG" }{TEXT -1 25 ". Test the proce
dure for " }{XPPEDIT 18 0 "n = 10,20,30;" "6%/%\"nG\"#5\"#?\"#I" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 257 "" 0 "" 
{TEXT -1 7 "- End -" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 1 "\004" }{MPLTEXT 1 0 0 "" }}}}{MARK "1 24 0" 99 }{VIEWOPTS 
0 0 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }