Summary: For any set of positive and relatively prime integers $A$, the set of positive integers that are not representable as a nonnegative integral linear combination of elements of $A$ is always a non-empty finite set. Thus we may define $g(A), n(A), s(A)$ to denote the largest integer in, the number of integers in, and the sum of integers in this finite set, respectively. We determine $g(A), n(A), s(A)$ when $A = {a, b, c}$ with $a | lcm(b, c)$. A particular case of this is when $A = {kl, lm, mk}$, with $k, l, m$ pairwise coprime. We also solve a related problem when $a | lcm(b, c)$, thereby providing another proof of the formula for $g(A)$.

11D07

linear Diophantine equation, Frobenius problem