Question

Write whether the square of any positive integer can be of the form $3m+2$, where $\mathrm{m}$ is a natural number. Justify your answer.

Answer 1

Determine the square of any positive integer can be of the form $3m+2$, where $\mathrm{m}$ is a natural number.

According to Euclid's division lemma, two positive integers $a$ and $b$, then there exist unique integers $q$and $r$ which satisfy the condition $a=bq+r$ where $0\le r<b$.

For $b=3$

$a=3\left(q\right)+r,$ where, $r$ can be integers,

For $r=0,1,2$

$3q+0,3q+1,3q+2$ are positive integers,

$$\begin{gathered} {\left(3q\right)}^2=9q^2 \\ \Rightarrow =3\left(3q^2\right) \\ \Rightarrow =3m \end{gathered}$$ (where $3q^2=m$)

$$\begin{gathered} {(3q+1)}^2={(3q+1)}^2 \\ \Rightarrow =9q^2+1+6q \\ \Rightarrow =3(3q^2+2q)+1 \\ \Rightarrow =3m+1\mathbf{}\mathbf{(}\mathit{W}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{,}\mathbf{}\mathit{m}\mathbf{}\mathbf{=}\mathbf{}\mathbf{3}{\mathit{q}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{q}\mathbf{)} \\ {(3q+2)}^2={(3q+2)}^2 \\ \Rightarrow =9q^2+4+12q \\ \Rightarrow =3(3q^2+4q+1)+1 \\ \Rightarrow =3m+1\mathbf{}\mathbf{(}\mathit{W}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{,}\mathbf{}\mathit{m}\mathbf{}\mathbf{=}\mathbf{}\mathbf{3}{\mathit{q}}^{\mathbf{2}}\mathbf{+}\mathbf{4}\mathit{q}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{} \end{gathered}$$

Hence, there is no positive integer whose square can be written in the form $3\mathrm{m}+2$ where $\mathrm{m}$ is a natural number.