Question

prove that n square minus n is divisible by 2 for any positive integer n

Answer 1

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

Answer 2

this is the easiest method