Math 310: Problem set 6

Instructions: This problem set is due Thursday, October 5, 2006. Your goal is not only to give correct answers but to communicate your ideas well. Make sure you use good English.
  1. Read 5ABC.
  2. Optional: Read the New Yorker article on events concerning Perelman's sketched solution of the Poincare Conjecture. One of the authors is Sylvia Nasar who wrote the book about John Nash that the movie of the same name, "A Beautiful Mind", is based on. (Also of interest is the reply of Yau's lawyer.)
  3. Let b and c be positive integers. If gcd(b,c)=8, what are the possible values of gcd(b3,c4)? Give examples of specific values of b and c that show each value of gcd(b3,c4) that you claim is possible really is possible, and justify why those are the only possible values of gcd(b3,c4).
  4. Let m, b and c be positive integers. Prove that [mb,mc] = m[b,c].
  5. Let b and c be positive integers. Show that the smallest positive integer k such that b divides kc is k = b/gcd(b,c).
  6. Suppose b and c are positive integers such that gcd(b,c)=12 and [b,c]=120 with b <= c (where I use "<=" to mean "less than or equal to", since it's not easy to use the standard symbol in a way that any web browser will display correctly). Find all possible values of b and c. Justify your answer.
  7. Suppose m, x and y are positive integers. If x == y (mod m), prove that gcd(m,x) = gcd(m,y).