Assume all variables are integers greater than 1.
Suppose $y = k x$ and $k \neq x$. Suppose that $z\lt k$ and $\gcd(k, z) \gt 1$.
Does this imply $\gcd(z, x) \lt x$?
Assume all variables are integers greater than 1.
Suppose $y = k x$ and $k \neq x$. Suppose that $z\lt k$ and $\gcd(k, z) \gt 1$.
Does this imply $\gcd(z, x) \lt x$?
No. Take for example $k=8$, $x=2$, and $z=4$.