How to prove that $p$ does not divide $2^{p} - 1$?

Question

(Note that this question is an edited version of a previous question, which I cannot delete because answers have been submitted.)

Suppose $p$ is an odd prime.

How does one prove that $p$ does not divide $2^{p} - 1$?

Answer 1

Since $p$ is odd it is not divisible by $2$ and so by transitivity it is not divisible by $2^{p-1}$.

Answer 2

it is $$2^{p-1}$$ and even number (product of factors $2$) and $$p$$ is an odd number.

Answer 3

Hint (now you've finally stabilised the question): Fermat's little theorem tells you that $p$ does divide $2^p - 2$.