Prove that the square of any positive integer of the form 5q + 1 is of the same form.

To Prove: that the square of a positive integer of the form 5q + 1 is of the same form

Proof: Since positive integer n is of the form 5q + 1 

If n = 5q + 1

Then $n^{2}=(5 q+1)^{2}$

$\Rightarrow n^{2}=(5 q)^{2}+(1)^{2}+2(5 q)(1)$

$\Rightarrow n^{2}=25 q^{2}+1+10 q$

$\Rightarrow n^{2}=25 q^{2}+10 q+1$

$\Rightarrow n^{2}=5\left(5 q^{2}+2 q\right)+1$

$\Rightarrow n^{2}=5 m+1$ (where $m=\left(5 q^{2}+2 q\right)$ )

Hence $n^{2}$ integer is of the form $5 m+1$