Math:abstract algebra problem

This is the problem i'm having trouble proving:

Let a,b be integers such that gcd(a,b)=1. Let r,s be integers such that ar+bs=1 prove that gcd(ab)=gcd(r,b)=gcd(r,s)=1.

We already know that gcd(a,b) = 1. We need to show that gcd(r,b) = 1, and that gcd(r,s) = 1.

Let d be a common divisor of r and b. Then d divides ar and bs, so d divides ar + bs = 1. So d = +-1. So the only common divisors of r and b are +-1. So gcd(r,b) = 1. The argument that gcd(r,s) = 1 is exactly the same ((try it!)).

I hope this helps!