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title: MTH 350 Homework 6
tags: [' homework', mth350]

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tags: mth350, homework
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# MTH 350 Homework 6

Instructions for this Homework set are on Blackboard where this is posted. The due date is on the class calendar, which is linked on Blackboard. 



## Problem 1

:::warning
This problem is an **independent** problem. As described in [the Syllabus](https://docs.google.com/document/d/1Rest_DodWnDy7Y8EVhp2lEOVnWcTEjLAGfWV2khuQQY/edit?usp=sharing) (top of page 14) this means that **on this problem, the only help you can get is from the professor.** 
:::

Suppose $a,m \in \mathbb{N}$ and $a,m > 1$. Prove that if $\gcd(a,m) > 1$, then $\overline{a}$ is a zero divisor in $\mathbb{Z}_m$. 

**Notes and hints:**
- **Do not start this proof until you have practiced and are clear on the concept of zero divisors.**. Suggestion: For $m = 2, 3, \dots, 20$, find all the zero divisors of $\mathbb{Z}_m$, make sure you understand why they are zero divisors, and check this with your group (or a computer). Example: $\overline{6}$ is a zero divisor in $\mathbb{Z}_9$ because $\overline{6} \cdot \overline{3} = \overline{0}$. 
- This is based on work we started in class on February 22. Feel free to use any part of that discussion; but if a proof was only partial, make sure to give a complete proof in your submission. 
- Without loss of generality you may assume $a < m$ since this proof has to do with the congruence class of $a$ in $\mathbb{Z}_m$, which we always represent with an integer less than $m$. 
- Because we are assuming $1 < a < m$ it means that you can assume $\overline{a} \neq \overline{0}$. That's important for proving $\overline{a}$ is a zero divisor! 
- Remember that $\overline{a}$ is a zero divisor if $\overline{a} \neq \overline{0}$ (which we just said we can assume) and there exists a class $\overline{b}$ *which is also not equal to $\overline{0}$* such that $\overline{a} \cdot \overline{b} = \overline{0}$. That part about $\overline{b}$ not equalling $\overline{0}$ is essential, so make sure it's part of your proof. 



## Problem 2

:::success
This problem is a **collaborative** problem. This means that **you can work on it in a small group, as long as you stay within the bounds of academic honesty** laid out in [the Syllabus](https://docs.google.com/document/d/1Rest_DodWnDy7Y8EVhp2lEOVnWcTEjLAGfWV2khuQQY/edit?usp=sharing) (starting on page 13). 
:::

<!-- **Choose exactly one of the following statements** and write a complete, correct, and clear proof. -->
<!-- 
You can (and should) test-drive each of these problems in your notes to see which one feels best for you, but please only turn in work on one of these (in addition to Problem 1).  -->

1. Prove the converse of Problem 1: If $\overline{a}$ is a zero divisor in $\mathbb{Z}_m$, then $\gcd(a,m) > 1$. (Use all the same assumptions that were given in Problem 1.) 
2. Prove that if $p$ is prime, then for all $\overline{x} \in \mathbb{Z}_p$ not equal to $\overline{0}$, the equation $\overline{x} \cdot \overline{y} = \overline{1}$ has a solution. 

**Notes and hints:**

- For option 1, you might test-drive different techniques for proving this "if-then" statement: Direct proof, contradiction, and proving the contrapositive. 


**Additional options for Problem 2 may be added through Tuesday, depending on class activities.**

## Submission instructions

You are turning in two items: 

1. Both parts of Problem 1.
2. Your choice of proof in Problem 2. 

Remember to type up your work using $\LaTeX$, and create a single PDF document with each problem on a separate page. (Use the command `\pagebreak` to start a new page.) 

Then upload your PDF to the "Homework 5" assignment area, and remember to hit **Submit**. 

**Handy $\LaTeX$ commands:** 

- To put a bar over the top of something, enclose it in `\overline{}` in math mode. For example `$\overline{x}$` creates $\overline{x}$; `$\overline{x^2 + y^2}$` creates $\overline{x^2 + y^2}$. 
- To get the "integers mod $m$" just use the fancy $\mathbb{Z}$ for integers and subscript it with $m$. For example `$\mathbb{Z}_6$` gives $\mathbb{Z}_6$, and `$\mathbb{Z}_{100}$` gives $\mathbb{Z}_{100}$. Or if you are using the macro `$\Z$` defined in the template, it's just `$\Z_6$` or `$\Z_{100}$` 
- To typeset a basic integer congruence like $10 \equiv 2 \pmod 8$, use `\equiv` to get the triple-equals sign, and `\pmod` to get the "mod". For example, `$10 \equiv 2 \pmod 8$` gives $10 \equiv 2 \pmod 8$. 