Suppose $a$ and $b$ are relatively prime and $x \in \mathbb{Z}$. Show that if $a|bx$, them $a|x$ [duplicate]
Solution 1:
Your second solution is great, and you can state that $a$ cannot divide $b$ because of the reasons above.
One tiny thing I would suggest you add, depending on how harsh the graders might be (if this were graded), would be to state why $sx$ and $tq$ are each integers, and why $k$ is therefore an integer.