Use Euclid’s division lemma to show that the square of any positive integer is of the form 3p, 3p + 1 or 3p + 2.

Let be any positive integer. Then, it is form 3q or, 3q + 1 or, 3q + 2

So, we have the following cases:

Case I When a = 3q

In this case, we have

a² = (3q)² = 9q² = 3q(3q) = 3p, where p = 3q²

Case II When a = 3q + 1

In this case, we have

a² = (3q + 1)² = 9q² + 6q + 1 = 3q(3q + 2) + 1 = 3p + 1,

where p = q(3q + 2)

Case III When a = 3q + 2

In this case, we have

a² = (3q + 2)² = 9q² + 12q + 4 = 9q² + 12q + 3 + 1

= 3(3q² + 4q + 1) + 1 = 3p + 1

where p = 3q² + 4² + 1

Hence, a is the form of 3p or 3p + 1 or 3p + 2