I want to prove that doesn't exist $k<5^{n}-5^{n-1}$ where $5^{n}\mid 2^{k}-1$.
I know that the Euler-Fermat Theorem says \begin{equation} 2^{\phi(5^{n})}\equiv 1 \mod 5^{n} \end{equation} where $\phi(5^{n})=5^{n}-5^{n-1}$. I wonder if this is the minimum exponent that accomplish this equation. If $k$ not divide $\phi(5^n)$ and $k$ is the least, is evident. But what occur if $k\mid \phi(5^n)$? I'm stuck.