【Solution】Solution: (1) If Factory B specializes in making coats for a month, it can complete:
$$
\begin{array}{l}
3600 \div\left(\frac{3}{2+3}\right), \\
=3600 \div \frac{3}{5} \\
=6000 \text { (pieces). }
\end{array}
$$
(2) If Factory A first specializes in making 6000 pairs of pants, it would take:
$$
\begin{aligned}
& (6000 \div 2700) \times \frac{1}{2+1}, \\
= & \frac{10}{3} \times \frac{1}{3}, \\
= & \frac{20}{27} \text { (months); }
\end{aligned}
$$

The remaining time Factory A can still produce clothing:
$$
\begin{array}{l}
2700 \times\left(1-\frac{20}{27}\right), \\
=2700 \times \frac{7}{27}, \\
=700 \text { (sets). }
\end{array}
$$
(3) If the two factories cooperate for a month, the maximum number of clothing sets they can produce is:
$$
6000+700=6700 \text { (sets). }
$$

Answer: If the two factories cooperate for a month, the maximum number of clothing sets they can produce is 6700. Therefore, the answer is: 6700.
17. (5 points) Factory A and Factory B produce the same type of clothing. Factory A produces 2700 sets of clothing per month, with the time ratio for producing tops and pants being 2:1; Factory B produces 3600 sets of clothing per month, with the time ratio for producing tops and pants being 3:2. If the two factories cooperate for one month, the maximum number of clothing sets they can produce is $\qquad$ sets.