4. (3 points) On the Island of Misfortune, there live knights, who always tell the truth, and liars, who always lie. One day, $n$ islanders gathered in a room. $\quad 30$

The first one said: "Exactly every first person present in this room is a liar."

The second one said: "Exactly every second person present in this room is a liar." and so on

The person with number $n$ said: "Exactly every $n$-th person present in this room is a liar." How many people could have been in the room?
# Answer: 2

## Solution:

Let's note that among these people, there is definitely one knight, otherwise it would mean that the first islander is a liar telling the truth, which is impossible. Moreover, the first person is definitely a liar.

Furthermore, there is exactly one knight, since all the speakers contradict each other. This means there are no more than half knights. On the other hand, all the islanders claim that there are no fewer than half knights. Therefore, there are exactly half, i.e., there are two people in the room.