Given integers $a, b$ with $b \neq 0$, suppose there is $n \geq 2$ such that $n |b$ but $n \nmid a$. Show $\frac{a}{b}$ is not an integer.
I'm not entirely certain about this but does supposing $n \geq 2, n |b$ but $n \nmid a$ tell us about $a, b$ being coprime? If so, the proof is easy enough to follow from there. Otherwise i'm not sure.
Hint Assume by contradiction that $\frac{a}{b}$ is an integer, call it $k$.
Then $a=bk$, or in other words $b|a$. You also know that $n|b$. What can you conclude from here?