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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 integerm? Justify your answer.


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Solution

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

Now, on squaring the number

3q+12=9q2+6q+13q+12=3(3q2+2q)+13q+12=3m+1

where m=3q2+1 is an integer.

No, its square cannot be written in the form of 3mor3m+2

Hence, 3q+12cannot be expressed in any other form apart from 3m+1


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