Prove that if n is a positive integer, then n is even if and only if 7n 4 is even.

We prove first the direct implication. Assume n is even. Then n = 2k for some integer k. Then 7n + 4 = 14k + 4 = 2 (7k + 2), which is even. For the converse, which is “if 7n + 4 is even, then n is even”, we use a proof by contrapositive. The contrapositive is: “if n is not even (that is, odd), then 7n + 4 is not even (that is, odd) If n is odd, then n = 2k + 1, for some integer k. Then 7n + 4 = 14k + 11 = 2 (7k + 5) + 1, which is odd.