---
tags: mth350, dailyprep
---

# Daily Prep 5B -- MTH 350-01

## Overview

We've introduced a major character in MTH 350, namely $\mathbb{Z}_n$, or the set of congruence classes of integers modulo $n$. We've seen that with "mod $n$" addition and multiplication, $\mathbb{Z}_n$ forms its own finite number system. In Module 5B we will begin to explore how arithmetic and algebra might work in this number system, with a particular focus on finding the **units** and the **zero divisors** in $\mathbb{Z}_n$.

## Learning objectives

**Basic Learning Objectives:** *Before* our class meeting, use the Resources listed below to learn all of the following. You should be reasonably fluent with all of these tasks prior to our meeting; we will field questions on these, but they will not be retaught.

- State the definition of a **zero divisor** in $\mathbb{Z}_n$ and give examples and nonexamples of zero divisors in $\mathbb{Z}_n$ for specific choices of $n$.
- State the definition of a **unit** in $\mathbb{Z}_n$ and give examples and nonexamples of units in $\mathbb{Z}_n$ for specific choices of $n$.
- State the definition of a **multiplicative inverse** of an element of $\mathbb{Z}_n$.

**Advanced Learning Objectives:** *During and after* our class meeting, we will work on learning the following. Fluency with these is not required prior to class.

- State and prove a conjecture that gives necessary and sufficient conditions for $[a] \in \mathbb{Z}_n$ to be a zero divisor.
- State and prove a conjecture that gives necessary and sufficient conditions for $[a] \in \mathbb{Z}_n$ to be a unit.
- Given a unit in $\mathbb{Z}_n$, find a multiplicative inverse for it.

## Resources for learning

**Reading:** Read through Investigation 5 (pp. 73--75) in the textbook. Skip or scan the discussion about binary operations and well-definedness.

**Video:** Here's a video (homemade!) with an example of finding the zero divisors and units in $\mathbb{Z}_6$:

<iframe src="https://player.vimeo.com/video/509978306" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe>

## Exercises

The exercises for this Daily Prep are found on the Google Form:

https://docs.google.com/forms/d/e/1FAIpQLSdcNZMT3frKuGpCON4tfatk-1Vdt1vUOQwh319fiBV9QGhuyw/viewform

## Submission and grading

To submit your work, simply submit the Google Form. You will receive a receipt via email to confirm your submission. (Look in your spam folders if you do not see the receipt.)

A **Pass** mark is given if the Daily Prep is turned in before its deadline and if each item on the Daily Prep has a response that represents a good faith effort to be right. **Mistakes are not penalized**. A **No Pass** is given if an item is left blank (even accidentally), has an answer but it shows insufficient effort (including responses like "I don't know"), or if the Daily Prep is late.