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.

True

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False

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Solution

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

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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.

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.