Example 4 Find the last two digits of the positive integer $\left[\frac{10^{93}}{10^{31}+3}\right]$.
Analysis Write the fraction $\frac{10^{93}}{10^{31}+3}$ in the form of a mixed number to make the solution of this problem easier.
Solution Since $10^{93}+3^{3}=\left(10^{31}\right)^{3}+3^{3}$
$$
=\left(10^{31}+3\right)\left(10^{62}-3 \times 10^{31}+9\right) \text {, }
$$

thus the original expression $=\left(10^{31}\right)^{2}-3 \times 10^{31}+9+\left[\frac{-3^{3}}{10^{31}+3}\right]$
$=10^{31} \times\left(10^{31}-3\right)+9-1$
$=10^{31} \times\left(10^{31}-3\right)+8$.

Therefore, the last two digits of the required number are 0 for the tens place and 8 for the units place.