**Given**

if n is odd then 8 divides n^{2}-1

**Proof**

We can proof the given statement directly.

Let n **∈** O

Let,

P = n^{2} – 1

We need to prove that P is divisible by 8

P = n^{2} – 1^{2}

P = (n + 1)(n – 1)

We know that n is odd. This implies that n + 1, n – 1 must be even. Let n – 1 = x

P = x * (x + 2)

Both x and x+2 are even. When we divide x by 2, we have 2 cases

**CASE 1: x / 2 = y, y is odd**

Divide P by 8

**CASE 2: x / 2 = y, y is even**

In both the cases, P is completely divisible by 8.

Hence proved.