Existence of Frobenius Numbers without using Schur's Theorem

Question

I learned about Frobenius problem today. I am confused about the existence of the Frobenius numbers.

It says according to Schur's theorem, the Frobenius number exists, but I want to know if it has an easier proof?

Answer 1

The case of two coprime coin values $a$ and $b$ can be regarded as a simple exercise in modular arithmetic.

Consider the set of values $S:=\{ka\}$ with $0\le k<b$. Then each of these values are distinct $\bmod b$, otherwise you would have $s_i-s_j\equiv 0 \bmod b$, giving $b \mid a(i{-}j)$ with $i{-}j<b$, which is not possible due to the coprime condition.

So $S$ forms a complete set of different residues $\bmod b$ and any larger value than $a(b{-}1)$ (the largest value in $S$) can be formed by adding a suitable multiple of $b$ to the corresponding equivalent value in $S$ that is equivalent $\bmod b$ to the target value.

The largest value that can't be represented by addition of non-negative multiples of $a$ and $b$ is thus $a(b-1) - b = ab - a -b$ which is nicely symmetric between the terms, as expected.

For more than two coins, the existence of a minimum value is easy enough - just use pairs of coins in succession to fill the residue list - but the minimum value is a little harder to identify due to possible interactions when the coins are only coprime as a complete set rather than pairwise (eg $\{6,10,15\}$).