1. Abstract- divisisbility

Show that if a | c and b | c, and (a, b) = d, then ab | cd.

Abstract divisibility.
We have c=am, c=bn, and d=an+bm.

2. Could I multiply d=an+bm by c.
Then I have cd=acn+bcm
We have cd=c(an+bm)
cd=cd
Thus, ab | cd

3. Does that make sense or should I have:
cd=c(an+bm)
I think somehow I need to get c(an+bm) to be of the form abr, right?