Question

Prove that n²-n is divisible by 2 for every positive integer n

Answer 1

There's a more simple way to prove it

Casei: Let n be an even positive integer.
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: Let n be an odd positive integer.
When n = 2q + 1
In this case
n2 -n = (2q + 1)2 - (2q + 1)= (2q +1) ( 2q+1 -1)= 2q (2q + 1)
n2 - n = 2r , where r = q (2q + 1)
n2 - n is divisible by 2.

∴ n 2 - n is divisible by 2 for every integer n