Proving the Chinese Remainder Theorem for Reduced Residues Modulo mn

[frowdow]

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 $$\phi$$$$\left(xy\right)\neq$$$$\phi\left(x\right)\phi\left(y\right)$$

 

[fresh_42]

This description is a bit weird. Look up the chinese remainder theorem and the Euclidean algorithm. I guess that $n,m$ are coprime.