## Math 310: Problem set 9

Instructions: This problem set is due Friday, March 10, 2006.
1. 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. (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.
3. 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