I rename $s$ into $t$: Peter wrote: (Four Number Theorem) Let $a, b, c,$ and $d$ be positive integers such that $ab=cd$. Show that there exists positive integers $p, q, r,t$ such that \[a=pq, \;\; b=rt, \;\; c=pt, \;\; d=qr.\] Let $p=\gcd\left(a,\ c\right)$. Then, $p\mid a$ and $p\mid c$. Define $q=\frac{a}{p}$ and $t=\frac{c}{p}$; then, $q$ and $t$ are positive integers. Then, $a=pq$ and $c=pt$. Hence, $ab=cd$ becomes $pqb=ptd$, so that $qb=td$. The numbers $q$ and $t$ are coprime (in fact, $p=\gcd\left(a,\ c\right)=\gcd\left(pq,\ pt\right)=p\gcd\left(q,\ t\right)$ yields $\gcd\left(q,\ t\right)=1$). Now, $qb=td$ yields $t\mid qb$. Since $q$ and $t$ are coprime, this leads to $t\mid b$. Define $r=\frac{b}{t}$; then, $r$ is a positive integer, and $b=rt$. Finally, $ab=cd$ yields $d=\frac{ab}{c}=\frac{pq\cdot rt}{pt}=qr$. Thus, our four positive integers $p$, $q$, $r$, $t$ satisfy $a=pq, \;\; b=rt, \;\; c=pt, \;\; d=qr$. Hence, we are done. Darij

Corrected (at least in my post).

Peter wrote: (Four Number Theorem) Let $ a, b, c,$ and $ d$ be positive integers such that $ ab = cd$. Show that there exists positive integers $ p, q, r,s$ such that \[ a=pq,\;\; b=rs,\;\; c=ps,\;\; d=qr.\] We use the following observation. Fact. If $ q$ is a positive rational number, then we can write $ q=\frac{m}{n}$ for some positive integers with $ \gcd(m,n)=1$. Since $ \frac{a}{c}=\frac{d}{b}$, one can find $ q, s\in\mathbb{N}$ such that $ \gcd(q,s)=1$ and \[ \frac{a}{c}=\frac{d}{b}=\frac{q}{s}.\] Since $ \frac{a}{c}=\frac{q}{s}$ and since $ \gcd(q,s)=1$, we can write \[ a=qp,\; c=sp\] for some positive integer $ p$. Also, since $ \frac{d}{b}=\frac{q}{s}$ and since $ \gcd(q,s)=1$, we can write \[ d=qr,\; b=sr\] for some positive integer $ r$.