Midterm 1 Study Guide

Material covered

Up to and including chapter 6.

• Definitions to know:
  • Divisibility
  • gcd and lcm
  • prime and composite numbers
• You should know all theorem and lemma statements. Especially:
  • Well-ordering principle
  • Every number is a product of primes
  • Euclidean division
  • Euclidean algorithm
  • gcd is the smallest integer linear combination of the terms
  • Every number is a product of primes in a unique way (up to order of factors)
  • lcm divides all other common multiples
  • Applications of Fundamental Theorem
• Theorems you should know how to prove:
  • Sketch the idea of why every number is a product of primes
  • Prove that the gcd is the last non-zero remainder in the euclidean algorithm
  • Prove that lcm(a, b) divides all other common multiples of a, b
  • Prove the condition about when a diophantine equation \(ax+by=c\) has a solution.

Practice Problems

• Know very well all the turned-in assignments (1-5)
• Know how to do the non-optional practice problems
• Be ready for true/false questions
• Being able to do basic proofs by induction
• Solving diophantine equations