###### tags: `MATA02-2022`
# Ungraded problems 9

You are to work on these questions only for the benefit of learning the material. These problems are meant for practice, not an assignment to be handed in.

Fermat's little theorem: If $p$ is a prime and $a$ is nonzero on modulo $p$, then $a^{p-1} \equiv 1$ mod $p$.

## 1.
Use Fermat's little theorem to evaluate the power $11^{55}$ mod $19$.

## 2.
Use Fermat's little theorem to evaluate the power $2^{83}$ mod $23$.

## 3.
Use Fermat's little theorem to evaluate the power $3^{2022}$ mod $17$.

# Selected (partial) solutions
<details>
<summary>Solutions to selected problems (click to expand)</summary>

1. 11
2. 18
3. 15

</details>