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.

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Solution

## Given the positive integer $3q+1$ where $q$ is a natural number. Now, on squaring the number $⇒{\left(3q+1\right)}^{2}=9{q}^{2}+6q+1\phantom{\rule{0ex}{0ex}}⇒{\left(3q+1\right)}^{2}=3\left(3{q}^{2}+2q\right)+1\phantom{\rule{0ex}{0ex}}⇒{\left(3q+1\right)}^{2}=3m+1$ where $m=3{q}^{2}+1$ is an integer. No, its square cannot be written in the form of $3mor3m+2$Hence, ${\left(3q+1\right)}^{2}$cannot be expressed in any other form apart from $3m+1$

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