Using the Division Algorithm, prove that 3|(n3 - n) for every integer n. Proof: If n = 0, then n' - n = and thus 3| (n - n). If n = 1, then n - n = and thus 3| (n3 - n). If n = 2, then n' - n = and thus 3| (n) - n). It follows that n - n = 3k for some ke Z, if n = 0, 1 or 2. Suppose now that n E Z and n *. Then, by the Division Algorithm, n = , where q E Z and v Thus n - n = and since r = 0, 1 or 2, therefore re - r = for some ke Z. Thus n' - n = 3(9q + 9q'r + 3qr- - q + k), and so v Q. E. D. A. 27q' + 27q'r + 9gr? + 3 - (3q + r) E. 3|(n] - n) B. (3q + r) - (3q + r) FO