**Given**

n^{2} – 2 is not divisible by 4 ∀ n ∈ N

**Proof**

For value of n, we have 2 choice

**CASE 1 : n is even**

If n is even, then we can write n as

n = 2x, ∀ x ∈ N

n^{2} – 2 = 4x^{2} – 2

= 2( 2x^{2} – 1 )

2x^{2} is always even for ∀ x∈ N. Therefore, 2x^{2} – 1 is always odd.

Therefore, 2( 2x^{2} – 1 ) is not divisible by 4. We can say that n^{2} – 2 is not divisible by 4, ∀ n ∈ E

**CASE 2 : n is odd**

If n is even, then we can write n as

n = 2x – 1, ∀ x ∈ N

n^{2} – 2 = ( 2x – 1 )^{2} – 2

= 4x^{2} + 1 – 4x – 2

= 4x( x – 1 ) – 1

4x( x – 1 ) is always even for ∀ x∈ N. Therefore, 4x( x – 1 ) – 1 is always odd. Therefore, 4x( x – 1 ) – 1 is not divisible by 4. We can say that n^{2} – 2 is not divisible by 4, ∀ n ∈ O

From the above two cases, we conclude that n^{2} – 2 is not divisible by 4 ∀ n ∈ N.