Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.

To Prove: that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.

Proof: Since positive integer n is of the form of 2q or 2q + 1 

If n = 2q

Then, $n^{2}=(2 q)^{2}$

$\Rightarrow \quad n^{2}=4 q^{2}$

$\Rightarrow \quad n^{2}=4 m\left(\right.$ where $\left.m=q^{2}\right)$'

If n = 2q + 1

Then, $n^{2}=(2 q+1)^{2}$

$\Rightarrow n^{2}=(2 q)^{2}+4 q+1$

$\Rightarrow n^{2}=4 q^{2}+4 q+1$

$\Rightarrow n^{2}=4 q(q+1)+1$

$\Rightarrow n^{2}=4 q+1$ (where $m=q(q+1)$ )

Hence it is proved that the square of any positive integer is of the form 4q or 4q + 1, for some integer q.