Question

If n is an odd integer, then show that n² – 1 is divisible by 8.

Answer 1

We know that any odd positive integer n can be written in form 4q + 1 or 4q + 3.

When n = 4q + 1,

Then n² – 1 = (4q + 1)² – 1

= 16q² + 8q + 1 – 1

= 8q(2q + 1) is divisible by 8.

When n = 4q + 3

Then n² – 1 = (4q + 3)² – 1

= 16q² + 24q + 9 – 1

= 8(2q² + 3q + 1) is divisible by 8.

So, from the above equations, it is clear that

If n is an odd positive integer

n² – 1 is divisible by 8.

Hence Proved.