5. Given the sets $A=\{x \mid 5 x-a \leqslant 0\}, B=\{x \mid 6 x-b>0\}, a, b \in \mathbf{N}$, and $A \cap B \cap \mathbf{N}=\{2,3$, $4\}$, the number of integer pairs $(a, b)$ is $\qquad$.
5. 30 Detailed Explanation: $5 x-a \leqslant 0 \Rightarrow x \leqslant \frac{a}{5} ; 6 x-b>0 \Rightarrow x>\frac{b}{6}$, to make $A \cap B \cap \mathbf{N}=\{2,3,4\}$, then $\left\{\begin{array}{l}1 \leqslant \frac{b}{6}<2 \\ 4 \leqslant \frac{a}{5}<5\end{array}\right.$, which means $\left\{\begin{array}{l}6 \leqslant b<12 \\ 20 \leqslant a<25\end{array}\right.$, so the number of pairs $(a, b)$ is $C_{6}^{1} C_{5}^{1}=30$.