Problem 4. Ana can run a circular track three times in 8 minutes. Maria can run the same track twice in 5 minutes. If Ana and Maria start running at the same time from the start, what is the total number of laps they will run before they meet again at the start for the first time? How many times has each of them run the track when they meet again at the start for the first time?
Solution. In one minute, Ana will run $\frac{3}{8}$ of the track, and Maria will run $\frac{2}{5}$ of the track in one minute. For them to meet again at the starting point, each of them needs to run the track a whole number of times. Therefore, we need to determine the smallest possible number of minutes $k$ such that $\frac{3}{8} k$ and $\frac{2}{5} k$ are natural numbers. Clearly, $k=\mathrm{LCM}(5,8)=40$, which means the total number of laps they will run is $40\left(\frac{3}{8}+\frac{2}{5}\right)=40 \cdot \frac{31}{40}=31$. Thus, after 40 minutes, they will meet again at the starting point for the first time. Ana will run the track $\frac{3}{8} \cdot 40=15$ times, and Maria will run the track $\frac{2}{5} \cdot 40=16$ times.