Prove that the products of two consecutive positive integers is divisible by 2.

Let the numbers are a and a-1


Product of these number: a(a-1) = a2-a


Case 1: When a is even:


a=2p


then (2p)2 - 2p ⇒ 4p2 - 2p


2p(2p-1) ………. it is divisible by 2


Case 2: When a is odd:


a = 2p+1


then (2p+1)2 - (2p+1) ⇒ 4p2 + 4p + 1 - 2p – 1


= 4p2 + 2p ⇒ 2p(2p + 1) ………….. it is divisible by 2


Hence, we conclude that product of two consecutive integers is always divisible by 2


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