Answer     100

By Euler's theorem, we have $2^{100}\equiv 1\pmod{125}$. So it is sufficient to check all the divisors of $100$: $2$, $4$, $5$, $10$, $20$, $25$, and $50$. $$\begin{array}{rcll} 2^2&\equiv&4&\pmod{125} \\ 2^4&\equiv&16&\pmod{125} \\ 2^5&\equiv&32&\pmod{125} \\ 2^{10}&\equiv&24&\pmod{125} \\ 2^{20} &\equiv&76&\pmod{125} \\ 2^{25}&\equiv&57&\pmod{125} \\ 2^{50}&\equiv& -1 &\pmod{125} \end{array}$$

Therefore, we conclude the answer is $\boxed{100}$.