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

Question

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

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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 ?

Answer 1

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 $2^k = 2^n \equiv 1 (\mod 2^n-1)$