Math 310: Problem set 8

Instructions: This problem set is due Friday, March 3, 2006.
  1. (2 points) Find the least positive residue of 123473693458 mod 19.
  2. (2 points) For each prime p from 2 to 11, find the least positive residue of ip-1 mod p, for each i in the range 0 < i < p. (Make a table to make it easy to display your results.)
  3. (2 points) Make and clearly state a conjecture as to the value of ip-1 mod p when 0 < i < p and p is prime.
  4. (2 points) Do you think your conjecture also holds when i > p or i < 0? Explain why you think so or why not. (What about when p divides i?)
  5. (2 points) Explain how your conjecture could be used to simplify your computation of 123473693458 mod 19.
  6. (2 points) Find an integer solution x which is simultaneously congruent to 17 mod 35 and to 4 mod 99. Explain how you find x.