Daily Prep 2A -- MTH 350-01

--- tags: mth350, dailyprep --- # Daily Prep 2A -- MTH 350-01 ## Overview Now that we've established the basic axioms for integer arithmetic and ordering, we will take a deep look at **integer division**. We'll begin by defining what it means for one integer to divide another (without talking about fractions!) and move toward proving the **Division Algorithm**, a fundamental result in algebra and arithmetic which will be a dominant tool for a lot of our work later in the course. We'll eventually work up a proof of the Division Algorithm, which hinges on the **Well Ordering Principle**. As a side quest we will define **integer congruence modulo $n$** and prove some useful properties of integer congruence. ## 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 *divides* in the context of integers; use the definition to decide if one integer divides another and to rephrase a division relationship. + State the *Division Algorithm*; apply it to identify the quotient and remainder when dividing one integer by another. + State the *Well-Ordering Principle*. **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. + Write a proof of the Division Algorithm. + Explain line-by-line the proof of the Division Algorithm. ## Resources for learning **Reading:** Read through Investigation 2, just pages 11--17 in the textbook. **Video:** Watch the following videos, total running time 15:44. + Integer divisibility (from the MTH 210 playlist): https://youtu.be/dIfpZzX7bKo (8:53) + The Division Algorithm (also from MTH 210): https://youtu.be/XHjSy_MT7u0 (6:51) ## Exercises The exercises for this Daily Prep are found on the Google Form: https://docs.google.com/forms/d/e/1FAIpQLSe9tXLSyJV3FD6zWZS2Y3EcbsBT0DTJ38EHdW0AOXQdjymPKQ/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.