Question

Say true or false:

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

• A. True
• B. False

Answer 1

Answer: (B)

Let the positive integer $n$ is of the form $3q,3q+1,$ and $3q+2$
If $n=3q$
Squaring both sides, we get,
$=>n^2=9q^2$
$=>n^2=3\left({3q}^2\right)$
$=>n^2=3m$, where $m=3q^2$
Now, if $n=3q+1$
$=>n^2={(3q+1)}^2$
$=>n^2=9q^2+6q+1$
$=>n^2=3q(3q+2)+1$
$=>n^2=3m+1,$ where $m=q(3q+2)$
Now, if $n=3q+2$
$=>n^2={(3q+2)}^2$
$=>n^2=9q^2+12q+4$
$=>n^2=3q(3q+4)+4$
$=>n^2=3q(3q+4)+3+1$
$=>n^2=3m+1$ where $m=(3q^2+4q+1)$
Hence, $n^2$ integer is of the form $3m$ and $3m+1$ not $3m+2$