4. In a school, there are 920 students, and in each class, there are no more than $k$ students. All students need to go on a bus tour. For this, 16 buses, each with 71 seats, have been booked. The students need to be seated on the buses so that students from each class end up in the same bus. For what largest $k$ is this guaranteed to be possible?
Answer: 17.

Solution. If $k=18$, then it is not always possible to seat the students in the buses. Indeed, suppose the school has 50 classes of 18 students each and two classes of 10 students each. Only three classes of 18 students can fit in one bus. Therefore, it would require at least 17 buses to transport students from such classes.

We will show that if there are no more than 17 students in each class, it is always possible to seat the students in the buses. Number the seats in the buses consecutively: from 1 to 71 in the first bus, from 72 to 142 in the second, and so on. Seat the students in the order of increasing seat numbers, starting with all students from the first class, then from the second, and so on. As a result, the students will fit into 13 buses, since $13 \cdot 71=923>920$. With this seating arrangement, students from no more than 12 classes can end up in two buses. It is sufficient to use three buses to accommodate these classes, since any four classes can fit in one bus.