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

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Solution

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

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