Elementary Number Theory Question with GCD If $n|at$, show that $\frac{n}{\gcd(a,n)}|t$.

I've been trying to figure this out for hours, even though it seems very basic (I think I'm missing something fundamental here). What would be the way to show this?

Let $d$ be the gcd of $a$ and $n$. Then $a=a_1d$ for some integer $a_1$, and $n=n_1d$ for some integer $n_1$, where $a_1$ and $n_1$ are relatively prime. 
We are told that $n_1d$ divides $a_1dt$. It follows that $n_1$ divides $a_1t$. Since $n_1$ and $a_1$ are relatively prime, we conclude that $n_1$ divides $t$.
