Let $q=\gcd(x,y)$. Prove that if $x\nmid zq$ then $x \nmid yz$.

Question

Let $x,y,z \in \mathbb{Z}$. Let $q=\gcd(x,y)$. Prove that if $x\nmid zq$ then $x \nmid yz$.

I understand that I can prove this using contrapositive, that is

Let $q=\gcd(x,y)$. If $x\mid yz$ then $x \mid zq$.

Assume $x|yz$ and $q = \gcd(x , y)$ then $q|x$ and $q|y$.

Then $xk = yz$ for some $k\in \mathbb{Z}$. $qr = x$ for some $r\in \mathbb{Z}$. $qs = y$ for some $s\in \mathbb{Z}$.

By Bezout's Theorum $q = xa + yb$ for some $a,b \in \mathbb{Z}$

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I'm not entirely sure how to proceed.

Answer 1

Ignoring the equations involving $r$ and $s$, we have everything we need. Observe that: \begin{align*} q &= xa + yb \\ zq &= xaz + (yz)b \\ zq &= xaz + (xk)b \\ zq &= x \cdot \underbrace{(az + kb)}_{\in ~ \mathbb Z} \end{align*}