On September 4, 1996, scientists used a supercomputer to find the 33rd Mersenne prime, which is the largest prime number known to humans so far. It is: $2^{125787}-1$ (378632 digits). Try to find the last two digits of this prime number.
Solution: First, find the unit digit.
$$
\begin{array}{l}
\because 2^{1257787}-1=2 \cdot 4^{628893}-1=8 \cdot 16^{31446}-1 \\
\quad \equiv 8 \times 6-1(\bmod 10) \equiv 7(\bmod 10),
\end{array}
$$
$\therefore 2^{1257787}-1$ has a unit digit of 7.
Next, find the tens digit.
$$
\begin{aligned}
\because & \frac{1}{10}\left[\left(2^{1257887}-1\right)-7\right] \\
= & \frac{1}{10}\left[2 \cdot 4^{628895}-8\right]=\frac{1}{5}\left(4^{628893}-4\right) \\
= & \frac{4}{5}\left(4^{628892}-1\right)=\frac{4}{5}\left(16^{314446}-1\right) \\
= & \frac{4}{5} \times(16-1) \times\left(16^{314445}+16^{314444}+\cdots+16\right. \\
& \quad+1) \\
\equiv & 12(6 \times 314445+1)(\bmod 10) \\
\equiv & 2(\bmod 10),
\end{aligned}
$$
$\therefore 2^{1257787}-1$ has a tens digit of 2.
Therefore, the last two digits of $2^{1257787}-1$ are 27.