homework2ans - MA 315 Spring 2011 Homework 2 Due Wednesday February 2 2011 Your paper must be typed Each proof must be written in complete sentences

MA 315, Spring 2011 Homework 2 Due Wednesday, February 2, 2011 Your paper must be typed. Each proof must be written in complete sentences with correct spelling, punctuation and grammar. (1) Prove the following: If n is an odd number then n 2 - 1 is divisible by 8. Proof: Let n be an odd integer. Then there exists an integer k such that n = 2 k + 1. Therefore n 2 - 1 = (2 k + 1) 2 - 1 = 4 k 2 + 4 k = 4 k ( k + 1). This previous line says that n 2 - 1 is divisible by 4, we now show that n 2 - 1 is actually divisible by 8. Note that k can be either even or odd; we will show that, in either case, n 2 - 1 is divisible by 8. If k is even, then k = 2 m for some integer m , therefore n 2 - 1 = 4 · 2 m ( k + 1) = 8 m ( k + 1), which shows that 8 | n 2 - 1, so n 2 - 1 is divisible by 8. If k is odd, then k +1 is even, so k +1 = 2 m for some integer m , so n 2 - 1 = 4 k · 2 m = 8 km which shows that 8 | n 2 - 1, so n 2 - 1 is divisible by 8. This yields that, whether k is even or odd, n 2 - 1 is divisible by 8.