2. In a company consisting of 2010 people, each person is either a knight, who always tells the truth, or a liar, who always lies. The people in the company are numbered from 1 to 2010. For each $k=1,2, \ldots, 2010$, the person with number $k$ said the following: "The number of liars among us is a divisor of $k$." How many liars can there be in such a company?
2. Solution. Let there be $m$ liars in this company. Then people with numbers $m, 2m, 3m, \ldots$ told the truth, while the rest lied. Therefore, the number of knights is exactly $\left[\frac{2010}{m}\right]$. We get the equation for $m: m + \left[\frac{2010}{m}\right] = 2010$. It is clear that both terms are integers, the product of which does not exceed 2010, so their sum can only be 2010 if one of the factors is 1. The case where $m=2009$ fits. Answer: there are 2009 liars in the company.