Find the last three digits of $383^{101}$

Answer: $\boxed{383}$. Let's solve it:

$\phi(125)=100$ and $\phi(8)=4$. Least common multiple of $100$ and $4$ is $100$.

$$383^{100} \equiv 1 \pmod{125}$$ and $$383^{4} \equiv 1 \pmod{8}$$ Therefore $383^{100} \equiv 1 \pmod{1000}$. Hence we yields $$383^{101} \equiv 383 \pmod{1000} $$