For any integer m, square of the number is of the form of 

Let a be any arbitrary number. 

Then, by Euclid’s division lemma, 

a = 3q + r where 0 ≤ r ≤ 3 

a²= (3q + r)² = 9q² + r² + 6qr 

When r = 0, 

Then, a² = 9q² + 0² + 6q(0) 

= 9q² = 3(3q²) = 3m where m = 3q² 

Now, r = 1 

a² = (3q+r)² 

= 9q² + (1)² + 6q(1) 

= 9q² + 1 + 6q 

= 3(3q² + 2q) + 1 = 3m + 1 where m = 3q²+ 2q 

When r = 2, 

a² = (3q+r)² 

= 9q² + (2)² + 6q(2) 

= 9q² + 12q + 4 

= 3(3q²) + 3(4)q + 3 × 1 + 1 

= 3[(3q²) + (4)q + 1] + 1 

= 3m + 1 where m = (3q²) + (4)q + 1 

Hence, square of number n is of the form 3m or 3m + 1