Say you have a 4x3 matrix P, whose entries are all integers.

What is a necessary and sufficient condition on $\displaystyle \mathbf b$ such that $\displaystyle P \mathbf x = \mathbf b$ (where $\displaystyle x = \begin{pmatrix} x_1 \\ x_2 \\ x_3\end{pmatrix}$ and $\displaystyle b = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \\ b_4 \end{pmatrix}$) has a solution?