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