2813 CSE Spring 2008 Quiz 4 (2/14/2008)
KEY
(8 point quiz--question 3 for a 1 point bonus)
1. [4 points] Let m be a positive integer. Show that ab(mod m) if a mod m = b mod m. We are given that a mod m = b mod m. This means that the remainder of a when divided by m is the same as the remainder of b when divided by m. So: a = q1m + r b = q2m + r a-b = q1m q2m = m(q1 q2) since q1 and q2 are integers q1 q2 is an integer. The definition of a b (mod m) is that a-b = km for some integer k. Since q1 q2 is an integer, we have shown that if a mod m = b mod m then a (mod b m) 2. [2 point each] Determine whether the integers in each of these sets are pairwise relatively prime. If they are not, explain why not. a) 12,17,31,37 Yes, these are pairwise relatively prime since there are no common factors among the four numbers. b) 14, 15, 21 No, these are not pairwise relatively prime: 14 and 21 share 7 as ...