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.

Solution:

No, its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m.

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 written in any other form apart from 3m + 1.

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