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## Subsection3.2Homework 1

Homework exercises are valuable preparation for your quiz on this material. Generally I do not collect homework for grading -- however, I will expect to review your homework progress in the event of your requesting a revision on a quiz. You are welcome to discuss homework problems with your groups and with the class, both in person and in the Slack #math channel.

### Subsubsection3.2.1Exercises

1. Exercises 8 #1.
2. Exercises 8 #3.
3. Exercises 8 #8.
4. Exercises 8 #13.
5. Exercises 8 #18.
6. Exercises 8 #19.
7. Exercises 8 #20.
8. Exercises 8 #25.
9. Exercises 8 #26.
10. Exercises 8 #28.
11. Prove using a direct proof: If there exists an integer $k$ such that $a = 6k + 5\text{,}$ then the equation $ax = 1$ (mod 6) can be solved for $x\text{.}$
12. Prove by contraposition or contradiction: If, for all $a$ satisfying $1 \leq a \leq m-1\text{,}$ the equation $ax \equiv 1 \quad \text{(mod m)}$ can be solved for $x\text{,}$ then $m$ is a prime number.