Math 311W Problem Set 5 Due Friday, October 16, 2015 1. Find the prime factorizations for 51 and 198, and use your answers to nd the gcd of 51 and 198. (No proof is required here, just write down the answers.) 2. Let m and n be positive integers. Prove that if m divides n, then (5m 1) divides (5n 1). 3. (a) Prove that if p is an odd prime and p > 3, then p is congruent to 1 or 5 mod 6. (b) Use (a) to prove that if n is congruent to 5 mod 6, then n has at least one prime divisor congruent to 5 mod 6. (c) Use (b) to prove that there are innitely many primes congruent to 5 mod 6. 4. Determine which of [3]48 , [7]48 , [11]48 , and [15]48 are invertible in Z48 , and nd the inverse of each invertible element. Which of these elements are zero-divisors? (Be sure to justify your answers.) 1