8. Find the last two digits of $2^{100}$.
Answer: 76
Solution: Note that $2^{12} \equiv 96(\bmod 100) \equiv-4(\bmod 100)$. Thus, $2^{100} \equiv\left(2^{12}\right)^{8}\left(2^{4}\right)(\bmod 100) \equiv(-4)^{8} 2^{4}$ $(\bmod 100) \equiv 2^{20}(\bmod 100) \equiv(-4) 2^{8}(\bmod 100) \equiv 76(\bmod 100)$