Can we deduce that $x$ is a common divisor of $y$, $z$ and $u$

Question

Let us consider the following relations between integers:

$x$ divides $yw$

$x$ divides $zt$

$x$ divides $uq$

and

$x$ is not a common divisor of $w$, $t$ and $q$. Here $w$, $t$ and $q$ are composite.

Can we deduce that $x$ is a common divisor of $y$, $z$ and $u$? If not, can we add some conditions to guarantee this property?

PS: We can consider $x$ as the product of three primes. We assume that those primes are factors of $w$, $t$, and $q$ respectively.

Answer 1

No. For example, take $x$ to be any product of three primes $abc$ and set $y=a,w=bc$ and $z=b,t=ac$, etc.)

I don't see a natural extra condition that forces this (other than $x$ being coprime to each of $w,t,q$ in which case it is not a very interesting statement). For example, you can certainly have $w,t,q$ being pairwise coprime without it working, by taking $x=abc,y=ab,z=bc,u=ac,w=cd,t=ae,q=bf$ where $a$ to $f$ are distinct primes.