Show that if $M$ can be written as the sum of squares of two integers, so can $2M, 5M, 8M, 10M, 13M$ and so on..
So I have figured out this question for the most part, if $M=a^2+b^2$ then I can use the Diophantus identity
where $c^2+d^2$ is the coefficient (2, 5, 8, etc).
However this is for a complex analysis course and he wants us to prove this 'by considering $[(a+bi)(c+di)]^2$ and I'm not sure where this comes into it.
My first thoughts were something along the lines of $a^2+b^2=(a+bi)(a-bi)$ or $a^2+b^2=[(ac-bd)+i(ad+bc)]^2$, but I haven't gotten anywhere with that.
Any help would be much appreciated, I don't need a complete solution, just a nudge in the right direction.