Deriving Largest Unattainable Score with GCD(m,n)=1

[imagination10]

Suppose you have a game in which there are two kinds of scoring events. One event gives a score of m points, and the other gives a score of n points. Assume gcd(m,n)=1. Derive a formula for the largest unattainable score. Prove your answer is correct.

Well, I only can figure of these:

since (m, n)= 1, then we have

{0, m, 2m,..., (n-1)m} as a complete residue system mod n

and

{0, n, 2n,..., (m-1)n} as a complete residue system mod m.

What should I do then?

 

[StatusX]

Start with am+bn=N, which has integer solutions a,b for all N since (m,n)=1. Now, these solutions aren't necessarily positive, but if a,b are one set of solutions, then a+kn, b-km are also solutions, where k ranges over the integers.

Now assume a<0, in which case b>N/n. Now try new solutions a₁=a+n, b₁=b-m. If a₁<0 still, then b₁>N/n, and we can continue this process until a_i>=0. But at this point, b_i will only remain positive if N is greater than... See if you can complete this argument.