MATH 319 Group Exercise #1 Maple and the GCD

Names_________________________________________________

1. Log on to the computer and open Maple 6 under Programs or Math Applications. Maple does exact arithmetic unless you tell it otherwise. In this first exercise we will just compute some numbers. Note that multiplication always requires a multiplication sign (*). Also, the semicolon is the end of line operator. The semicolon followed by enter will execute the operation. sqrt is the exact square root operator.

Compute the following numbers:

a) (2)(3)(5)(7)(11)+1=______________ b) = __________

b) 2¹² + 1 = ____________ d) 2⁵⁰ + 1 = ____________

e) = ____________ f) = ____________

2. Go to File, Open, Athena, MATH . Find the MATH Out Box and look for a program called GCD2 under my name. Load that program. Note that to assign a value to a variable one uses := (colon followed by equals).

The program calculates the value of the GCD of two numbers – a and b. It also at the bottom gives the numbers – x and y – such that GCD(a,b) = ax + by.

Find :

a) (2¹⁰-1,2¹²-1) = _________ b) (10⁹-1,37³) = ________

c) (2¹⁰⁰-1,2¹²⁰-1) = __________ d) (3⁵+2⁴,3⁷+2⁵) = ________

Find values of x and y in the integers that satisfy: (if possible)

a) 33x + 72y = 3, x = ____, y = _____, b) 33x + 72y = 12, x =____, y =_____

c) 54x + 86y = 3, x = ____, y = ____, d) 81x + 107y = 1, x = ______, y = _______

e) (2¹⁴+1)x + (3¹⁰ – 1)y = 1, x = _______________, y =_____________

3. In Maple there is a package of functions useful in Number Theory. To load these type with(numtheory); The function of interest is ifactor. This function writes out the prime factorization of a given number. For example ifactor(24); would yield (2³⁾(3).

On a new line define n = 10. (Remember the colon equals). We will check that the product of five consecutive integers is divisible by 120.

The product of (n-2)(n-1)n(n+1)(n+2) is n⁵ – 5n³ + 4n.

On the line below n:=10 type ifactor(n^5-5*n^3+4*n);

For each of the following values of n compute ifactor(n^5-5*n^3+4*n). Does it have factors that yield 120 as a factor?

n	ifactor(n^5-5*n^3+4*n)
10
20
100
10289

Note that you can change the value of n in the line above ifactor(n^5-5*n^3+4*n).