Answer :

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

a^{2} = (3q)^{2} = 9q^{2} = 3q(3q) = 3p, where p = 3q^{2}

__Case II When a = 3q + 1__

In this case, we have

a^{2} = (3q + 1)^{2} = 9q^{2} + 6q + 1 = 3q(3q + 2) + 1 = 3p + 1,

where p = q(3q + 2)

__Case III When a = 3q + 2__

In this case, we have

a^{2} = (3q + 2)^{2} = 9q^{2} + 12q + 4 = 9q^{2} + 12q + 3 + 1

= 3(3q^{2} + 4q + 1) + 1 = 3p + 1

where p = 3q^{2} + 4^{2} + 1

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

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