**Given**

If n is not divisible by 3 then n^{2}-1 is divisible by 3 ∀ n ∈ Z

**Proof**

For the value of n, we have 3 choices

**Case 1 : n % 3 = 1**

Let,

P = (n^{2}-1) % 3

P = ( ( n^{2} ) % 3 – 1 % 3 ) % 3

P = ( ( n % 3 ) ( n % 3 ) – 1 ) % 3

P = ( 1 * 1 – 1 ) % 3

P = 0 % 3

P = 0

Therefore, n^{2}-1 is divisible by 3 if n % 3 = 1

**Case 2 : n % 3 = 2**

Let,

P = (n^{2}-1) % 3

P = ( ( n^{2} ) % 3 – 1 % 3 ) % 3

P = ( ( n % 3 ) ( n % 3 ) – 1 ) % 3

P = ( 2 * 2 – 1 ) % 3

P = 3 % 3

P = 0

Therefore, n^{2}-1 is divisible by 3 if n % 3 = 2

**Case 2 : n % 3 = 0**

P = ( n^{2} – 1 ) % 3

P = ( ( n^{2} ) % 3 – 1 % 3 ) % 3

P = ( ( n % 3 ) ( n % 3 ) – 1 ) % 3

P = ( 0 * 0 – 1 ) % 3

P = -1 % 3

P = 2

Therefore, n^{2}+2 is not divisible by 3 if n % 3 = 0

From the above cases, we can conclude that If n is not divisible by 3 then n^{2}-1 is divisible by 3 ∀ n ∈ Z.