I need to show that:
If $a\mid b$ then $\gcd(a,c) \leq \gcd(b,c)$ where $a,b,c$ are positive integers.
I've come up with this, but I'm not 100% sure that it's correct:
Assume $a\mid b$, then $a \leq b$. Multiply both sides by an integer, $x$, such that $ax \leq bx$. Then add $cy$ for positive integer, $c$, and some integer, $y$, to both sides such that $ax + cy \leq bx + cy$. Then by EEA, $ax + cy = \gcd(a,c)$ and $bx + cy = \gcd(b,c)$. Therefore, $\gcd(a,c) \leq \gcd(b,c)$