Q. 33.9( 67 Votes )

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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


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


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