Question

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

Answer 1

Here's your answer:-

Any positive integer is in the form 2q or 2q+1 ; where q is some integer.

When n=2q

n^2+n=(2q)^2 +2q

=4q^2+2q

= 2q(2q+1)

Which is divisible by 2

When n=2q+1

n^2-n=(2q+1)^2+(2q+1)

=4q^2+4q+1+2q+1

=4q^2+6q+2

=2q(2q^2+3q+1)

Which is divisible by 2

Hence n^2+n is divisible by 2