Prove: $n\in\mathbb{N}$ is not divisible by $2^n-1$ if $n>1$

Asked Nov 26 '15 at 19:25

Active Nov 26 '15 at 19:25

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Using induction:

For $n=2\Rightarrow 2$ is not divisible by $3$

For $n=k\Rightarrow k$ is not divisible by $2^k-1$

For $n=k+1\Rightarrow k+1$ is not divisible by $2^{k+1}-1$

How to prove this by induction?

asked Nov 26 '15 at 19:25

user300045

6

Try proving that $2^n-1 > n$ if $n>1$ – Mathmo123 Nov 26 '15 at 19:28
Try using the contraposition: $k+1$ is divisible by $2^{k+1}−1 \Rightarrow k$ is divisible by $2^k−1$ – hachemy Nov 26 '15 at 19:42

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