Question

Prove that n² – n divisible by 2 for every positive integer n

Answer 1

To prove n² – n divisible by 2 for every positive integer n.

We know that any positive integer is of the form 2q or 2q + 1, for some integer q.

So, following cases arise:

Case I :

When n = 2q.

In this case, we have

n² – n = (2q)² – 2q = 4q² – 2q = 2q(2q – 1)

⇒ n² – n = 2r where r = q(2q – 1)

⇒ n² – n is divisible by 2.

Case II :

When n = 2q + 1.

In this case, we have

n² – n = (2q + 1)² – (2q + 1)

= (2q + 1)(2q + 1 – 1) = (2q + 1)2q

⇒ n² – n = 2r where r = q (2q + 1)

⇒ n² – n is divisible by 2.

Hence n² – n is divisible by 2 for every positive integer n.