MTH 350 Homework 3

--- tags: mth350, homework --- # MTH 350 Homework 3 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.** ::: Let $a,b \in \mathbb{N}$. Prove that if $a | b$, then $a \leq b$. Notes on this: - This was a "fact" that we brought up in class while discussing the definition of GCD. It allows us to conclude that the "$e$" mentioned in part 2 of the definition really is less than or equal to $d$. You're being asked to provide a proof of this "fact". - One way to prove that $x \leq y$ is to show that $0 \leq y - x$. If you can prove the second statement, then you can just add $x$ to both sides of the inequality. ## 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. Let $a,b \in \mathbb{Z}$. Prove that if $a | b$ and $b | a$, then $|a| = |b|$.  2. Let $a \in \mathbb{Z}$. Prove that if $3$ does *not* divide $a$, then $3$ divides $a^2 - 1$. (Hint: If $3$ doesn't divide $a$, what does that imply about the remainder you get when dividing $a$ by $3$?) **Additional options for Problem 2 may be added through Tuesday, depending on class activities. Watch your announcements!** ## "Problem 3" This is not a mathematics problem, but a task to perform with your working groups. **Please schedule a one-hour meeting with your working group on or before Thursday February 3**. There is no class meeting on February 3, so although you can meet whenever you want, you are guaranteed to be able to use the February 3 class meeting time for this. The agenda for your meeting is: - **Make sure everyone in the group has been introduced to each other** --- names, majors, and so on. - **Make a document that contains contact information for each person in the group.** This can be a Google Doc or Spreadsheet, or something else. Just make sure each person has access to it in case you need to get in touch with each other. - **Agree on a way to get in contact regularly with each other to discuss questions and assignments.** This could be a weekly check-in meeting, or a group text, GroupMe/Telegram/etc., a list of emails, or something else, or a combination of these. I will not be checking to make sure you are meeting with each other regularly. But, the purpose of your group is to be able to reach out to each other to collaborate and give/receive help. Decide in your group how you'd like to do this and set it up. - **Spend at least 30 minutes working on Problem 2 in this Homework Set and/or the Daily Prep for Tuesday February 8**. These are items where collaboration is allowed, so make use of your group to do it! For "Problem 3" on this homework submission, please give a brief report about how your meeting went, and make a note of anything you're excited about, any concerns you have, and any questions that came up. ## Submission instructions You are turning in two items: 1. The proof for Problem 1. 2. Your choice of proof in Problem 2. 3. The report on your group meeting for "Problem 3". 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 3" assignment area, and remember to hit **Submit**.