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

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Let two consecutive positive integers is n, and (n + 1)

Product of both integers = n(n + 1) = n2 + n

We know that any positive integer is in the form 2q

and 2q + 1. where q is an integer.

Here two cases are possible

Case. I. when n = 2q then

⇒ n2 + n = (2q)2 + (2q)

⇒ n2 + n = 4q2 + 2q

⇒ n2 + n = 2q(2q + 1) [Let r = q(2q + 1)]

⇒ n2 + n = 2r

⇒ n2 + n, 2 can be divided by 2

⇒ n(n + 1), also divided by 2

So, product of two consecutive positive integer is divided by 2

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