Main Question or Discussion Point

I am teaching myself number theory using George Andrews book. i am stuck in the following problem:

To prove that as x cycles thru' the reduced residue set modulo m & y cycles thru' reduced residue set modulo n, nx + my cycles thru' reduced residue set modulo mn.

I am able to prove that:
a) nx1 + my1 [STRIKE]=[/STRIKE] nx2 + my2 and
b) nx + my is relatively prime to mn

But how do i prove that nx + my cycles thru' every one of the reduced residue set modulo mn without first proving that $$\phi$$$$\left(xy\right)\neq$$$$\phi\left(x\right)\phi\left(y\right)$$