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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 $\gcd(z, x) \lt x$?

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    @Arturo Magidin Thanks for the edit.2012-02-11

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No. Take for example $k=8$, $x=2$, and $z=4$.