3. A hundred people are sitting in a circle. Some of them are knights, who always tell the truth, and the rest are liars, who always lie. For some natural number $k<100$, each of those sitting said: "The next $k$ people sitting to my right are liars." What could the number $k$ be? (S. Berlov)
Answer. Any divisor of the number 100, except 1, decreased by $1: 1,3,4,9,19,24,49$, 99. Solution. If all the people sitting were liars, they would all tell the truth. Therefore, there must be a knight among them. After him, there are $k$ liars. The one sitting next to the knight, the next $k-1$ people are liars, and since he is lying, the $k$-th person after him is a knight. Reasoning this way further, we can see that between two consecutive knights at the table, there are $k$ liars. Thus, all the people sitting at the table are divided into groups of $k+1$ people, and therefore 100 must be divisible by $k+1$. Conversely, if 100 is divisible by $k+1$, the seating arrangement of people at the table in groups of a knight and $k$ liars following him satisfies the conditions of the problem.