Give a proof by contrapositive of If n3 is even then is even

Give a proof by contrapositive of: \"If n^3 is even then is even.

Solution

Proof by Contrapositive:
(If n is odd, then n^3 is odd.)

Suppose that n is odd. Writing n = 2k + 1 for some integer k,
n^3 = 8k^3 + 12k^2 + 6k + 1 = 2(4k^3 + 6k^2 + 3k) + 1.

Since 4k^3 + 6k^2 + 3k is an integer, we conclude that n^3 is odd.

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