Integer Properties Proofs

1. In the expressions below, n is an integer. Indicate whether each expression has a value that is an odd integer or an even integer. Use the definitions of even and odd to justify your answer. You can assume that the sum, difference, or product of two integers is also an integer. (a) 2n + 4 (b) 4n+3 (c) 10n3 + 8n – 4 (d) -2n2 – 5 2. Prove each statement using a proof by exhaustion. (a) For every integer n such that 0 ≤ n n3. (b) For every integer n such that 0 ≤ n 3n. 3. Find a counterexample to show that each of the statements is false. (a) Every month of the year has 30 or 31 days. (b) If n is an integer and n2 is divisible by 4, then n is divisible by 4. (c) For every positive integer x, x3 < 2x. 4. Use the given equations in a complete proof of the theorem. Your proof should be expressed in complete English sentences. (a) Theorem: If a, b, and c are integers such that a3|b and b2|c, then a6|c. b=ka3 c=jb2 c=jb2=j(ka3)2=(jk2)a6 5. Find the mistake in the proof. Explain where the proof uses invalid reasoning or skips essential steps. Theorem: If n and m are odd integers, then n2 + m2 is even (a) Proof. m = 7 is odd because 7 = 2·3+1. n = 9 is odd because 9 = 2·4 + 1. 72+92=49+81=130=2⋅65 2 2 Since 7 + 9 is equal to 2 times an integer, 72 + 92 is even. Therefore, the theorem is true.