Show that if $m^2 + n^2 $ is divisible by $4$, then $mn$ is also divisible by $4$. Show that if $m$ and $n$ are integers such that $m^2 + n^2 $ is divisible by $4$, then $mn$ is also divisible by $4$.
I am not sure where to begin. 

Note that $n^2$ is even if and only if $n$ is even.
Since $m^2+n^2$ is even, you see that either both $m,n$ are even, or both are odd.
If both $m$, $n$ are even, then $mn$ is divisible by $4$.
If both $m$, $n$ are odd, you have $m=2k+1$, $n=2l+1$ so $m^2+n^2=4k^2+4k+4l^2+4l+2$ is not divisible by $4$. This gives the result.
