Find the order of $2$ in $\mod 2^{n} -1 $ Find the order of $2$ in  $\mod 2^n-1$ 

I know that the order of $2$ in $\mod  2^n-1$ is the smallest positive integer $k$ such that
$$2^k  \equiv 1 \pmod {2^n-1}$$
How to proceed from here ? Any help/hints ?

Since you we use $\mod 2^n-1$ it is clear that $k$ must be greater than $n-1$ because $2^{n-1}\leq 2^n-1$ If you try $k=n$ you see that $2^k = 2^n \equiv 1 (\mod 2^n-1)$
