State true or false:The square of any positive integer is of the form 3m, 3m+1 but not of the form 3m+2.

# State true or false:The square of any positive integer is of the form but not of the form $3m+2$.

1. A
True
2. B
False

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### Solution:

We have to find whether the square of any positive integer is of the form but not of the form $3m+2$.
Euclid's Division Lemma states that, if two positive integers “a” and “b”, then there exists unique integers “q” and “r” such that which satisfies the condition:
$a=\mathit{bq}+r$ where  $0\le r\le b$
Consider ‘n’ as a positive integer for some integer q.
From Euclid's Division Lemma for $b=3$,
It can be of the form .
Substitute $n=3q$ and square on both sides,
${n}^{2}={\left(3q\right)}^{2}$ $⇒{n}^{2}=9{q}^{2}$ $⇒{n}^{2}=3\left(3{q}^{2}\right)$ Let $3{q}^{2}$ be ‘x’, then $n=3x$ where x is some integer.
Now substitute n = 3q+1 and square on both sides,
${n}^{2}={\left(3q+1\right)}^{2}$ $⇒{n}^{2}=9{q}^{2}+6q+1$ $⇒{n}^{2}=3\left(3{q}^{2}+2q\right)$ Let $\left(3{q}^{2}+2q\right)$ be ‘x’, then $n=3x+1$ where x is some integer.
Substitute n = 3q+2 in and square on both sides,
${n}^{2}={\left(3q+2\right)}^{2}$ $⇒{n}^{2}=9{q}^{2}+12q+4$ $⇒{n}^{2}=3\left(3{q}^{2}+4q+1\right)+1$ Let $\left(3{q}^{2}+4q+1\right)$ be ‘x’, $n=3x+1$ where x is some integer.
Thus, the square of any positive integer is of form but not of form $3m+2$.
Therefore, it is true that the square of any positive integer is of the form but not of the form $3m+2$.
Hence, option 1 is correct.

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