Answer:
To find the remainder when $100^{100}$ is divided by $18$, we can use the following steps:
1. Find the remainder of $100$ when divided by $18$, which is $4$.
2. Raise $4$ to the power of $100$.
3. Find the remainder of the result when divided by $18$.
Using modular arithmetic, we can calculate:
$100^{100} \equiv 4^{100} \equiv (4^2)^{50} \equiv 16^{50} \equiv (-2)^{50} \equiv 4 \pmod{18}$
So, the remainder when $100^{100}$ is divided by $18$ is $\boxed{4}$.
Note: The modular arithmetic steps use the fact that $16 \equiv -2 \pmod{18}$ and $(-2)^2 \equiv 4 \pmod{18}$.
