## Math 310: Problem set 9

*Instructions*: This problem set is due
Friday, March 10, 2006.
- Let a and b be integers, and let m and n be positive integers. Consider the problem:
x = a mod m
x = b mod n

- (a) (2 points) Give a specific example of integers a, b, m and n
such that the problem above has no solution. Explain what it is about your
example that guarantees there is no solution.
- (b) (2 points) Give a specific example of integers m and n
such that the problem above has a solution for every choice of a and b.
Explain what it is about your
example that guarantees there is always a solution.
- (c) (2 points) Give a specific example of integers a and b
such that the problem above has a solution for every choice of m and n.
Explain what it is about your
example that guarantees there is always a solution.

- (2 points) Find all integer solutions x which are simultaneously congruent to
17 mod 105 and to 5 mod 99. Explain how you find them.
- Find a solution to each of the following problems,
or explain why there is no solution.
- (a) (2 points) 5x = 1 mod 347
- (b) (2 points) 6x = 10 mod 15
- (c) (2 points) 6x = 9 mod 15