【Analysis】As shown in the following diagram, the Lica diagram
It is found that when A is at point A and B is at point B, the distance between the two is the greatest.
The time taken for the first maximum distance between the two is $\frac{36}{k}$ hours.
Thereafter, every $\frac{72}{k}$ hours, the two are at their maximum distance again.
Therefore, $p=\frac{72}{k} \times 2011+\frac{36}{k}$, $q=\frac{72}{k} \times 2012+\frac{36}{k}$,
Thus, $q-p=\frac{72}{k}$,
Hence, the maximum value of $k$ is 72
3、A and B are 36 kilometers apart. Supermen A and B start walking from $\mathbf{A}$ to $\mathbf{B}$ at the same time. Once they reach $\mathbf{B}$, they immediately walk back to $\mathbf{A}$, and upon reaching $\mathbf{A}$, they immediately walk back to $\mathbf{B}$...... They keep moving back and forth between $\mathbf{A}$ and $\mathbf{B}$. If Super A's speed is $2 k$ kilometers/hour, and Super B's speed is $k$ kilometers/hour. Suppose after $p$ hours, the distance between Super A and Super B reaches its maximum for the 2012th time; after $q$ hours, the distance between Super A and Super B reaches its maximum for the 2013th time. If $q-p$ is a positive integer, find: the maximum value of the positive integer $k$