Let $a$ and $b$ be coprime integers. Is it possible that $a \mid b$?
My thinking is that if $a \mid b$ then $a$ and $b$ share a factor besides $\pm 1$ ($a$ itself) and so are not coprime. Thus, $a \nmid b$.
This is probably very simple, but I'm still unsure.
Assuming $a$ and $b$ are positive:
If $a\mid b$, then, since $a\mid a$, $\gcd(a,b)\geqslant a$. So, unless $a=1$, no, you cannot have both $\gcd(a,b)=1$ and $a\mid b$.