$[a]_m \subseteq [a]_n$

Question

I am completely lost. I have tried using an element $x$ in $[a]_m$ such that $x \equiv a \pmod{n}$, and I know this means $x=a+nk$ for some integer $k$, but I do not know how to show this is a subset of $[a]_n$ or how to show $n|m$ from this. Please help!

Answer 1

We have $$ [a]_m=\{\ldots,a-m,a,a+m,a+2m,\ldots\} $$ and similarly for $[a]_n$. If we have $[a]_m\subseteq [a]_n$, then in particular $a+m\in[a]_n$. Which is to say there is some integer $k$ such that $a+m=a+kn$. It follows almost immediately that $n\mid m$.

Answer 2

Hint: $[a]_n=a+n\Bbb Z$. And, $m\Bbb Z\subset n\Bbb Z\iff n\mid m$.