3. Xiao Li and Xiao Zhang are running at a constant speed on a circular track. They start from the same place at the same time. Xiao Li runs clockwise and completes a lap every 72 seconds, while Xiao Zhang runs counterclockwise and completes a lap every 80 seconds. At the start, Xiao Li has a relay baton, and each time they meet, the baton is passed from one to the other (the time taken for passing the baton is negligible). The relay baton returns to the starting point in the minimum of $\qquad$ seconds.
3. 720 .

The time taken from the start to the first meeting is
$$
\frac{1}{\frac{1}{72}+\frac{1}{80}}=\frac{720}{19}
$$
seconds, during which Xiao Li runs $\frac{10}{19}$ laps, and Xiao Zhang runs $\frac{9}{19}$ laps.
Divide the circular track into 19 equal parts, and number the points clockwise from the starting point as $0,1, \cdots, 18$. Then,
from $0 \sim \frac{720}{19}$ seconds, the baton moves from 0 to 10;
from $\frac{720}{19} \sim \frac{720}{19} \times 2$ seconds, the baton moves from 10 to 1;
from $\frac{720}{19} \times 2 \sim \frac{720}{19} \times 3$ seconds, the baton moves from 1 to 11;
generally, from $\frac{720}{19} \times 2 k \sim \frac{720}{19} \times(2 k+1)$ seconds, the baton moves from $k$ to $10+k$; from $\frac{720}{19} \times(2 k+1) \sim \frac{720}{19} \times(2 k+2)$ seconds, the baton moves from $10+k$ to $1+k$.
Therefore, when $\frac{720}{19} \times(2 \times 9+1)=720$ seconds, the baton returns to the starting point for the first time.