COT3100Hmk05Sol - COT 3100 Fall 2002 Homework#5 Solutions 1 a Use Euclid's Algorithm to find the greatest common divisor of 962 and 629 962 = 1*629

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COT 3100 Fall 2002 Homework #5 Solutions 1) a) Use Euclid's Algorithm to find the greatest common divisor of 962 and 629. 962 = 1*629 + 333 629 = 1*333 + 296 333 = 1*296 + 37 296 = 8*37 GCD(962, 629) = 37 b) Use the Extended Euclidean Algorithm to find integers x and y such that 962x+629y = gcd(962, 629). 333 - 296 = 37 629 - 333 = 296 962 - 629 = 333 (962 - 629) - (629 - 333) = 37 962 + 333 - 2*629 = 37 962 + (962 - 629) - 2*629 = 37 2*962 - 3*629 = 37 x = 2, y=-3 is a solution. 2) Let x and y be integers. If 13 | (2x+5y), prove there are no integer solutions to 3x+y = 2003. Since 13 | (2x+5y), let 2x+5y = 13a, where a Z. 3x + y = 13(x+2y) - 5(2x+5y) = 13(x+2y) - 5(13a) = 13(x + 2y - 5a) Thus, we can conclude, since x,y and a are integers, that 13 | (3x + y). But, we see that 13 | 2003 is false. Thus, it is impossible for any integer solutions to exist to the given equation. 3) Given the prime factorization of n is p a q b , where p and q are prime and a and b are positive integers, determine the number of factors of n. (Hint: Note that each factor of n MUST only contain p and q in its prime factorization, and can not contain more than a p's or b q's.) As an example, find the total number of factors of 108 and list each of these. This is a counting question. ..) Each factor of n must be of the form p x q y , where x and y are non-negative integers with x a, and y b. Imagine making a table of all factors. Each row could be labelled by a value of x, and each column by a value of y.
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There are a+1 possible values of x (0, 1, 2, . .., a) and b+1 possible values
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COT3100Hmk05Sol - COT 3100 Fall 2002 Homework#5 Solutions 1 a Use Euclid's Algorithm to find the greatest common divisor of 962 and 629 962 = 1*629

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