Row Reduction & Rank · Challenge problem

Let $A = \begin{pmatrix} 1 & 2 \\ 2 & 4 \\ 3 & 7 \end{pmatrix}$. For how many vectors $\mathbf b = (b_1, b_2, b_3)$ with each $b_i \in \{0, 1, 2, 3, 4\}$ does the system $A\mathbf x = \mathbf b$ have a solution?