Suppose, $a,b,x,y$ are nonzero integers such that $w = ax = by$. Then $ w = \pm\text{lcm}(a,b)\gcd(x,y)$.
Proof. If $a, b = \pm 1$, then $\gcd(x,y) = \pm x = \pm y = w$. Let that be a base case for induction on the number $n$ of first few primes involved in forming $a, b$, ie. ($n = 0$ here). Now assume it's true for the first $n$ primes ie. If $a,b = \pm $ some composition of the first $n$ primes and $x, y$ are any nonzero integers, then the theorem is true. Now introduce the next prime $p$. Any number composed of the first $n+1$ primes only can be written as $pz$ where $z$ is composed only of the first $n$ primes. Let $a,b$ be such numbers, then if $w = a'x = b'y = ap^{c}x = bp^{d}y$ and $d\geq c$ wlog, then $\dfrac{w}{p^c} = a x = b(p^{d-c}y)$ and note that $p^{d-c}$ must divide $x$ since $a$ is composed only of the first $n$ primes. Then by induction $\dfrac{w}{p^c} = \pm \text{lcm}(a,b) \gcd(x, p^{d-c} y) = \pm \text{lcm}(a,b) p^{d-c} \gcd(x,y)$ so $w = \pm \text{lcm}(a,b) p^d \gcd(x,y) = \pm \text{lcm}(ap^c, bp^d) \gcd(x,y) = \pm \text{lcm}(a', b')\gcd(x,y)$. $\square$
Is that right?
