---
tags: mth350, dailyprep
---

# Daily Prep 3A -- MTH 350-01

## Overview

In Module 3 we'll be taking a look at a simple concept that has an impact beyond what you might expect -- the idea of the **greatest commo n divisor** (GCD) of two integers. We all learned how to find the GCD of two integers in elementary school, but the simplicity of the concept makes it a core tool for algebra moving forward. We'll be looking at a wide variety of mathematical results that pertain to the GCD, especially **Bezout's Identity** which states that every pair of integers can be written in a "linear combination" that equals their GCD. Along the way we meet two important algorithms: The **Euclidean Algorithm** and the **Extended Euclidean Algorithm**.

## 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 and apply the definition of the *greatest common divisor* of two integers.
- State the steps of the Euclidean Algorithm and use the Euclidean Algorithm to find the GCD of two integers.
- Define the concept of a *linear combination* of two integers; compute a given linear combination of two integers.
- State the definition of what it means for two integers to be *relatively prime*.

<!--
SAVE THESE FOR 3B
- Explain line-by-line a proof of Theorem 3.10
- State and apply Theorem 3.4, Theorem 3.9 (Bezout's Identity), Theorem 3.10, Corollary 3.11, Theorem 3.14, and Theorem 3.15. -->

**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.

- Explain line-by-line the proof of Theorem 3.4.
- Explain line-by-line a proof of the Eucidean Algorithm.
- Use the *Extended Euclidean Algorithm* to write the GCD of two integers as a linear combination of those integers.

## Resources for learning

**Reading:** Read through the first part of Investigation 3 (pp. 31--39) in the textbook.

**Video:**

- How to Find the Greatest Common Divisor by Using the Euclidian [*sic*] Algorithm (4:09) https://www.youtube.com/watch?v=JUzYl1TYMcU
- Euclidean Algorithm (10:01) https://www.youtube.com/watch?v=cOwyHTiW4KE <-- This isn't necessary, but it's another look at the algorithm and also contains a proof.
- Bezout's Identity (7:41) https://www.youtube.com/watch?v=7I92alYuF2M <-- Also contains information about the Extended Euclidean Algorithm which is not covered by name in your textbook.
- Extended Euclidean Algorithm Example (14:49) https://www.youtube.com/watch?v=6KmhCKxFWOs

<!-- **Other:** -->

## Exercises

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

https://docs.google.com/forms/d/e/1FAIpQLSe604JahDb2hhctrcw7DpONyokLTIn8z31AvOvMVXFyURRrhQ/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.