Non negative solution to diophantine equation

Question

If I have an equation $$ a_1x_1+a_2x_2+...a_nx_n=c$$ where $a_i,c$ are non negative integers. Then under what conditions if any are $x_i$ also non negative integer solutions?

Answer 1

There exist no conditions under which your equation has only integer solutions. Note that if some $\vec{x}=(x_1,\dots, x_n)$ is a solution, then all vectors of the form $\vec{x}+t\vec{n}, \, t\in \mathbb{R}$ are also solutions, if only $\vec{n}\perp\vec{a}$.