Divisibility Introduction

• Introduction and syllabus
• Introduction to CoCalc (one of your homeworks for Wednesday)
  • Accept your invitation
  • Look into the “Number Theory Project”, find the “SampleLaTeXPage folder” and go in it.
  • Click on the “SampleLaTeXPage.tex” file to view it. Do NOT edit it.
  • Source code on left, resulting PDF on right. Scroll through the source code and compare with the output in order to learn how to mark certain documents.
  • Open the project in a new window. Go into the “chapter 1” folder. Open the “SolutionsDay1.tex” file.
  • Add a new section with your name in the title. If you find other’s sections there, place yours relative to theirs according to the order of the problems you each have.
  • State the theorem or exercise you were assigned to do, then its solution.
• Definition: What are the natural numbers?
• Definition: What are the integers? How can we define the integers using the natural numbers?
• Definition: When do we say that an integer \(d\) divides another integer \(a\)? How do we denote it?
  • Provide examples.
• Think of edge cases:
  • Does an integer divide itself?
  • What integers does 1 divide? What integers divide 1?
  • What integers does 0 divide? What integers divide 0?
• Definition: When do we say that two integers \(a\), \(b\) are congruent modulo another integer \(d\)?
  • Provide examples.
• Think of edge cases:
  • When is an integer congruent to itself?
  • When is an integer congruent to 0?
  • When is an integer congruent to its negative?