Question

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

Answer 1

Since any positive integer $n$ is of the form $2m$ or, $2m+1$

If $n=2m,$ then
$n^2=4m^2=4q,$ where $q=m^2$

If $n=2m+1,$ then
$n^2={(2m+1)}^2=4m^2+4m+1=4m(m+1)+1=4q+1,$ where $q=m(m+1)$