Question

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

Answer 1

Let, (n – 1) and n be two consecutive positive integers
∴ Their product = n(n – 1)
= n ² −n
We know that any positive integer is of the form 2q or 2q + 1, for some integer q
When n =2q, we have
n ² − n = (2q)² − 2
= 4q² − 2q
2q(2q − 1)
Then n ² − n is divisible by 2.
When n = 2q + 1, we have
n​² − n = (2q + 1)² − (2q + 1)
= 4q² + 4q + 1 − 2q − 1
= 4q² + 2q
= 2q(2q + 1)
Then n​² − n is divisible by 2.
Hence the product of two consecutive positive integers is divisible by 2.