Question

A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.

Answer 1

No.

Justification:

Consider the positive integer 3q + 1, where q is a natural number.

(3q + 1)2 = 9q² + 6q + 1

= 3(3q² + 2q) + 1

= 3m + 1, (where m is an integer which is equal to 3q2 + 2q.

Thus (3q + 1)² cannot be expressed in any other form apart from 3m + 1.