Assume we have a integral vector $c\in \mathbb{Z}^n$ and an integer constant $b\in \mathbb{Z}$. Is there a necessary and sufficient condition for whether or not there exists a non-negative integer vector $x\in \mathbb{Z}^n_+$ such that $c^Tx=b$?
If so, if there is not such vector $x$, is there then a way to find a maximal $b' < b, b'\in \mathbb{Z}$ such that there exist such an $x$?
My motivation comes from a desire to strengthen constraints for MILP's.