Four Number Theorem : Let $a$, $b$, $c$, $d$ be integers such that $ab = cd$. Let $a$, $b$, $c$, $d$ be integers such that $ab = cd$. Then there exist integers $x$, $y$, $z$, $w$ such that $xy = a$, $zw = b$, $xz = c$, $yw = d$.
My Progress: I tried playing with $\gcd(a,c)$, $\gcd(a,d)$,$\gcd(b,c)$, $\gcd(b,d)$.
But I am not able to proceed.
Please post hints rather than solution, it helps me a lot.
Thanks in advance.

Here are some cases to consider:
If the products are equal to $0$, WLOG, if $a=0$, then $c$ or $d$ must be $0$.
If $a=0$ and $c=0$, let $x=0$. If $d=0$, then we let $y=0$ and choosing $w$ and $z$ should be easy. If $d\ ne 0$, we let $w=1$ and you can choose your $y$ and $z$ accordingly.
Now consider the cases where the product is non-zero.
$$\frac{a}{c}=\frac{d}{b}=\frac{y}{z}$$ where $y$ and $z$ are chosen to satisfy $\gcd(y,z)=1$. Try to argue how to determine $w$ and $x$ from here.
