How can I find multiple solutions for a system of equations?

Question

I'm writing a program for CheckIO.org that is supposed to return an array, $$ \begin{bmatrix} x\\ y\\ z \end {bmatrix} $$ , that satisfies the System of Equations $$ A \begin{bmatrix} x\\ y\\ z \end {bmatrix} = \begin{bmatrix} 0\\ 225\\ 315 \end {bmatrix} $$ (where $A$ is a given matrix), where all elements of 'xyz' are integers. I've tried Gaussian elimination and Cramer's rule, but each method only gives one answer and the elements are floats. I also tried $\vec{x} = \vec{b}*A^{-1}$, but that returned the wrong answer.