Given that $\gcd(a,b) = 1$ and $\gcd(c,d) = 1$, show that $\gcd(ac,bd) = \gcd(a,d) * \gcd(b,c)$.
The work that I have done so far goes as follows.
We write $$\gcd(ac,bd) = acv + bdu$$ then $$\gcd(a,d) = ax + dy\quad\text{and}\quad\gcd(b,c) = bs + ct$$ and so \begin{align} \gcd(a,d) * gcd(b,c) &= (ax + dy) * (bs + ct) \\ &= axbs + axct + dybs + dyct \end{align} I tried to factor out some terms, and get $ax(bs+ct) + dy(bs+ct)$ but then I am stuck here. I don't know what else to use to prove the answer.