Let A be an abelian group, written additively, and let $n$ be a positive integer such that $nx=0$ for all $x\in A$. Assume that we can write $n=rs$, where $r,s$ are positive relative prime integers. Let $A_r$ consist of all $x\in A$ such that $rx=0$, and similarly $A_s$ consist of all $x\in A$ such that $sx=0$. Show that every element $a\in A$ can be written uniquely in the form $a=b+c$ with $b\in A_r$, and $c\in A_s$.
I defined $f:A_r \times A_s \rightarrow A$ by $(b,c) \mapsto b+c$, I showed that $A_r, A_s$ are subgroups of $A$ and that this map defines an isomorphism.
Can you guys think of another (elementary, please) way of going about this problem? Any suggestions would be appreciated.