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

## Write the square root of $3q+1$ in form other than $3m+1$.The given positive integer is: $3q+1$Square and further simplify the given expression:${\left(3q+1\right)}^{2}=9{q}^{2}+1+6q\phantom{\rule{0ex}{0ex}}=3\left(3{q}^{2}+2q\right)+1...\left(i\right)$Take $m=3{q}^{2}+2q$,Now substitute the value of $m$ in the equation $\left(i\right)$:$3\left(3{q}^{2}+2q\right)+1=3m+1$Since, $q$ is a natural number.Hence, $m=3{q}^{2}+2q$ would definitely be some integer. Therefore, ${\left(3q+1\right)}^{2}$ cannot be written in any form other than $3m+1$.

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