One of $2^1-1,2^2-1,...,2^n-1$ is divisible by $n$ for odd $n$ 
Let $n$ be an odd integer greater than 1. Show that one of the numbers $2^1-1,2^2-1,...,2^n-1$ is divisible by $n$.

I know that pigeonhole principle would be helpful, but how should I apply it? Thanks.

Hints:
$$(1)\;\;\;2^k-1=2^{k-1}+2^{k-2}+\ldots+2+1\;,\;\;k\in\Bbb N$$
$$(2)\;\;\;\text{Every natural number $\,n\,$ can be uniquely written in base two }$$
$$\text{with maximal power of two less than $\,n\,$ }$$
