Let n be an integer. Prove that n3 is odd if and only if n is odd.
(←) Assume n is odd. This means that n = 2x + 1 for some integer x. Then n = (2x + 1)3 = 8x3 + 12.x2 + 6x + 1 = 2(4x3 + 6x2 + 3x) + 1 is odd.
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