Find the last two digits of $1032^{1032}$.
(Chinese Theorem)

According to the Chinese Remainder Theorem, we calculate $1032^{1032}$ modulo 4, then modulo 25. First, it is clear that $1032^{1032} \equiv 0 \bmod 4$. Next,

$$
1032^{1032} \equiv 7^{1032} \equiv(-1)^{516} \equiv 1 \quad \bmod 25
$$

From this, we easily deduce that $1032^{1032} \equiv 76 \bmod 100$.