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Find the remainder when $100^{100}$ is divided by $18$.

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tiwarishivansh8840

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 $(-2)^2 \equiv 4 \pmod{18}$.

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