Question

show that n² -1 is divisible by 8, if nis an positive odd integer

Answer 1

We know that any odd positive integer is of the form 4q + 1 or 4q + 3 for some integer q.

So,

Case I :

When n = 4q + 1

n² – 1 = (4q + 1)² – 1 = 16q² + 1 + 8q – 1

= 8q (2q – 1)

⇒ n² – 1 is divisible by 8

Case II :

When n = 4q + 3

n² – 1 = (4q + 3)² – 1 = 16q² + 9 + 24q – 1

= 8 (2q² + 3q + 1)

⇒ n² – 1 is divisible by 8

Hence n² – 1 is divisible by 8 for an odd positive integer n.