$A$ is local and $A^n$ and $A^m$ are isomorphic as $A$-modules.

Question

I need to show that $A^n$ and $A^m$ are isomorphic if and only if $n=m$, where $A$ is a local ring ($A$ is a commutative ring with unity). Then need to generalize this to any commutative local ring $A$. I can see how to generalize, I need to use the fact that every ring has a maximal ideal. But don't know how to prove that $n=m$ is necessary. I appreciate it.