## Math 310: Problem set 8

*Instructions*: This problem set is due
Friday, March 3, 2006.
- (2 points) Find the least positive residue of 12347369
^{3458}
mod 19.
- (2 points) For each prime p from 2 to 11, find the least positive residue
of i
^{p-1} mod p, for each i in the range 0 < i < p.
(Make a table to make it easy to display your results.)
- (2 points) Make and clearly state a conjecture as to the value of i
^{p-1} mod p
when 0 < i < p and p is prime.
- (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?)
- (2 points) Explain how your conjecture could be used to simplify
your computation of 12347369
^{3458} mod 19.
- (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.