China Mathematical Competition

Since $5x-a\le 0\Rightarrow x\le \frac{a}{5}$, $6x-b>0\Rightarrow x>\frac{b}{6}$. In order to satisfy $A\cap B\cap \mathbb{N}=\{2,3,4\}$, we have $$\{\begin{array}{l}1\le \frac{b}{6}<2,\\ 4\le \frac{a}{5}<5,\end{array}$$ or $$\{\begin{array}{l}6\le b<12,\\ 20\le a<25.\end{array}$$ So the number of pairs $(a,b)$ is $\left(\genfrac{}{}{0ex}{}{6}{1}\right)\cdot \left(\genfrac{}{}{0ex}{}{5}{1}\right)=30$. Answer: C.