MATH 135 Winter 2017: Assignment 7 Due at 8:25 a.m. on Wednesday, March 8, 2017 It is important that you read the assignment submission instructions and suggestions available on LEARN. 1. Prove there does not exist an integer x such that 6 | (x2 5). 2. Let n Z and n 1000. Prove that 8 | n if and only if the integer formed by the last (least significant) four digits of n is divisible by eight. 3. Let S = {[12], [3]1 , [13][4]} and T = {[616 ], [24] + [67], [158]} be subsets of Z17 . Prove S = T . 4. (a) Prove that f : Z Z99 defined by f (x) = [50][x] is not injective. (b) Prove that f : Z99 Z99 defined by f ([x]) = [50][x] is injective. A notation error was corrected on March 4 at 8:00 a.m. - f ([x]) replaced f (x). 5. Alice and Bob play a game starting with a pile of n 1 sticks. Each player on his or her turn can remove 1, 2 or 3 sticks from the pile. The last player to remove a stick wins. Alice goes first. Prove by induction that if n 0 (mod 4), then there exists a strategy that Bob can follow which guarantees he will win no matter what moves Alice makes. 5) 6. What is the remainder when 3(4 is divided by 7? Show your work. 7. Let p and q be primes. (a) Prove that p = q or gcd(p, p + q) = 1. (b) Prove that p + q = (p q)3 if and only if p = 5 and q = 3. Hint: Work modulo p +