There is a proof at the bottom of page 1, I understand it up until the point where it says "It follows from Theorem 4.7 of Rosen that the entries in such a row form a complete residue system modulo n. Thus, exactly $\phi(n)$ of them will be relatively prime to $n$, and thus relatively prime to $mn$." I don't know where the theorem is, but would appreciate if someone could explain why an element would have to be in one of the $\phi(n)$ rows such that it is coprime with $mn$.
http://gauss.math.luc.edu/greicius/Math201/Fall2012/Lectures/euler-phi.article.pdf