Rixita $\star \star$ Find the unit digit of $\left[\frac{10^{20000}}{10^{100}+3}\right]$.
Since $\frac{3^{200}}{10^{100}+3}<1, I=\frac{10^{20000}-3^{200}}{10^{100}+3}$ is an integer, so $\left[\frac{10^{20000}}{10^{100}+3}\right]=I$. Also, $I \equiv \frac{-3^{200}}{3} \equiv-3^{199} \equiv-3^{3} \cdot\left(3^{4}\right)^{49} \equiv-27 \equiv 3(\bmod 10)$, hence the unit digit of $\left[\frac{10^{20} 000}{10^{100}+3}\right]$ is 3.