I want to use Mathematica to solve the problem:

Find the maximum $k$ such that $6x+9y+20z=k$ does not have a non-negative solution.

I tried FrobeniusSolve. But what is the elegant way to find the maximum?

I know the theoretical background of this problem, and I know other ways of getting the solution. But I want to see how this can be done elegantly in Mathematica.