3 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 $(\quad)$.
(A) 20
(B) 25
(C) 30
(D) 42
Solve $5 x-a \leqslant 0 \Rightarrow x \leqslant \frac{a}{5} ; 6 x-b>0 \Rightarrow x>\frac{b}{6}$. 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 $\left\{\begin{array}{l}6 \leqslant b<12, \\ 20 \leqslant a<25 .\end{array}\right.$. Therefore, the number of pairs $(a, b)$ is $\mathrm{C}_{6} \mathrm{C}_{5}^{1}=30$. Hence, the answer is $\mathrm{C}$.