For any odd number, $n$, does there always exist $k$ such that $2^k-1$ is a multiple of $n$?

Question

I assume the statement is true, and I'm sure there may substantial evidence for is (such as it being a sequence on the OEIS). However I'm unsure if this is a well-known theorem (solved or not), or of the triviality of a proof.

I don't necessarily need to prove the statement, I just need to know the validity of it, (but you get brownie points if you can prove it, or hint towards a proof, of course).

Answer 1

The answer is "yes". This follows immediately from Euler's theorem : $$2^{\varphi(n)}\equiv 1\mod n$$ where $\varphi(n)$ is the totient function , holds for every odd $n$. Hence you can use $k=\varphi(n)$ to get a solution.