Homework for Week 2

Part 1

  1. Prove that there is no smallest positive rational number.
  2. Prove that n is odd if and only if n2 is odd.
  3. Prove that 1+2+22+23+24+...+2n=2n+1-1.
  4. Prove that for all a except a=1, the fraction (1-an+1)/(1-a) equals the sum 1+a+a2+...+an.
  5. Prove that if you divide 10n by 9 then the remainder is 1.

Part 2

  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 1: Due Tuesday 1/31
    • Part 2: Due Friday 2/3