For a two-digit natural number $AB$
$$\begin{aligned}
& p: A + B = 5 \\
& q: A \cdot B = 6
\end{aligned}$$
Given these propositions.
If the proposition $(\mathbf{p} \vee \mathbf{q}) \Rightarrow (\mathbf{p} \wedge \mathbf{q})$ is false, what is the sum of the possible values of the number $AB$?
A) 106
B) 125
C) 144
D) 163
E) 182