The problem is:
Extend the definition of gcd(a, b) to all pairs of integers (a, b) != (0, 0) by defining gcd(a, b) = gcd(|a|, |b|). Let r, s, t, u be integers such that ru st = 1. For any integers a and b, not both zero, show that gcd(ra + sb, ta + ub) = gcd(a, b).

What I've done so far:
ru st = 1 means gcd(r, s) = gcd(r, t) = gcd(u, s) = gcd(u, t) = 1.
We also know that gcd(a, b) = a*x1 + b*x2, x1 & x2 integers. Don't really know if that helps...
I've proved that these are all true: t|b & u|a & r|b & s|a. I guess it helps, but can't see how.
I'm stuck now, need a little push... Any help is appreciated.