**Let $N$ be an even number not divisible by 10. What is the tens digit of $N^{20}$? What is the hundreds digit of $N^{200}$?**
Parse $N=10 k \pm 2$ or $10 k \pm 4$.
$$
\begin{array}{c}
(10 k \pm 2)^{20} \equiv 2^{20}=1024^{2}=24^{2}=576(\bmod 100), \\
(10 k \pm 4)^{20} \equiv 4^{20}=1024^{4} \equiv 24^{4} \equiv 76^{2} \equiv 24^{2} \equiv 76(\bmod 100),
\end{array}
$$

Therefore, the tens digit of $N^{20}$ is 7.
$$
\begin{aligned}
(10 k \pm 2)^{200} & \equiv 2^{200}=1024^{20}=24^{20}(\bmod 1000), \\
24^{20} & =(25-1)^{20} \equiv 1(\bmod 125) .
\end{aligned}
$$

Among the three-digit numbers, only $376=3 \times 125+1$ leaves a remainder of 1 when divided by 125, and is divisible by 8. Therefore,
$$
(10 k \pm 2)^{200} \equiv 24^{20} \equiv 376(\bmod 1000) .
$$

Similarly, $(10 k \pm 4)^{200} \equiv 4^{200} \equiv 24^{40} \equiv 376(\bmod 1000)$.
Therefore, the hundreds digit of $N^{200}$ is 3.