$5 \nmid 2^{n}-1$ when $n$ is odd I want to prove that $$5 \nmid 2^{n}-1$$ where $n$ is odd.
I used Fermat's little theorem, which says $2^4 \equiv 1 \pmod 5$, because $n$ is odd then $4 \nmid n$ , so it is done.
can you check it and say that my proof is right or wrong.
thanks.

${\rm mod}\ 5\!:\,\ 2^{1+2j}\equiv 2(4^j)\equiv 2(-1)^j\equiv \pm2 \not\equiv 1$
