Unformatted text preview: Proof. Suppose that x and y are two consecutive integers. We may assume that x is the smaller of the two numbers. Thus y = x + 1. It follows that the product of x and y is equal to x 2 + x . Now, we consider two cases: when x is even, and when x is odd. If x is even, then choose a ∈ Z such that x = 2 a . It follows that: x 2 + x = (2 a ) 2 + 2 a = 4 a 2 + 2 a = 2(2 a 2 + a ) . Hence x 2 + x is even. In the other case, x is odd. Choose b ∈ Z such that x = 2 b + 1. It follows that: x 2 + x = (2 b + 1) 2 + (2 b + 1) = 4 b 2 + 4 b + 1 + 2 b + 1 = 2(2 b 2 + 3 b + 1) . Hence x 2 + x is even. In both cases, we ﬁnd that the product of consecutive integers is even. / 1...
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 Fall '08
 Weissman
 Math, Pythagorean Theorem, Mathematical proof, Martin H. Weissman

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