How to prove that d|c in gcd(ac,b) = gcd(c,b) if gcd(a,b) = 1?

[PsychonautQQ]

Homework Statement

if gcd(a,b) = 1, show that gcd(ac,b) = gcd(c,b)

Homework Equations

gcd(x,y) = xm + yn for integers n and m

The Attempt at a Solution

ax + by = 1
gcd(ac,b) = d
gcd(c,b) = g

ac = dr
b = ds
c = gm
b = gn

I've been setting up equations and rearranging things but can't make any leeway, any tips?

Update:
g|c and g|b, so g|ac and hence g|d.

d|ac and d|b. If I can show that d|c then i can conclude d|g. hence d=g. I now need help showing that d|c.

 

[RUber]

If d is gcd(ac,b) then d|ac and d|b. If d >1, then d cannot divide both a and b, so d must divide c.