Question

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

Answer 1

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

Case 1: When n = 2 q 

n² – n = (2q)² – 2q = 4q² – 2q 

= 2q (2q – 1) 

In n² – n = 2r 

2r = 2q(2q – 1) 

r = q(2q + 1) 

n² – n is divisible by 2

Case 2: When n = 2q + 1 

n² – n = (2q + 1)² – (2q + 1) 

= 4q² + 1 + 4q – 2q – 1 = 4q² + 2q 

= 2q (2q + 1) 

If n² – n = 2r

r = q (2q + 1) 

∴ n² – n is divisible by 2 for every positive integer “n”