Homework for Week 2

Part 1

  1. When a=24, b=15, find gcd(a, b) and express it as a sum as+bt for some integers s,t.
  2. When a=114, b=153, find gcd(a,b) and express it as a sum as+bt for some integers s,t.
  3. Using Bezout's theorem, find integers s and t so that 5s+13t=1. Prove that these integers are not unique.
  4. True/False. If true, prove; if false, explain why (eg, with a counterexample). gcd(n,n+1)=1 for all positive integers n.
  5. True/False. If true, prove; if false, explain why (eg, with a counterexample). gcd(n,n+2)=1 for all positive integers n.
  6. What is gcd(2n+1, 5n+6)? (Explain/prove your answer.)

Part 2

Problems 10-14 from Fundamental Theorem of Arithmetic Lecture.