# Number Theory - Largest Impossible Score

1. Feb 25, 2007

### Frillth

Edit: I finally figured this problem out, although I didn't use residue systems, so a solution is no longer necessary.

1. The problem statement, all variables and given/known data

For this problem, I need to find a fomula for the largest unattainable score in a game where you can either score m or n points at a time (m and n are relatively prime) and prove that this formula will always work. Complete residue systems are suggested to be used in the proof.

2. Relevant equations

{0, n, 2n, 3n, ... , (m-1)n} is a complete residue system mod m when m and n are relatively prime.

3. The attempt at a solution

Through trial and error, I have found that the largest impossible score is mn - (m+n). I'm stuck, however, on proving this. I don't even know how to start, so any help would be greatly appreciated.

Last edited: Feb 25, 2007