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

Case I: Let n be an even positive integer.

Then, n = 2q

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.

Then, n = 2q + 1

⇒ n² –n

= (2q + 1)² – (2q + 1)

Taking (2q+1) common from both the terms

= (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, Hence proved