11.5. On the coordinate plane, the graph of $y=\frac{2020}{x}$ is constructed. How many points on the graph have a tangent that intersects both coordinate axes at points with integer coordinates?
Answer: 40 points. Solution. The equation of the tangent at the point $\left(x_{0}, y_{0}\right)$ to the hyperbola $y=k / x$ is $y-y_{0}=-\left(k / x_{0}^{2}\right)\left(x-x_{0}\right)$, where $y_{0}=k / x_{0}$. From this equation, the coordinates $x_{1}$ and $y_{1}$ of the points of intersection with the axes O $x$ and O $y$ are obtained, specifically, $x_{1}=2 x_{0}$ and $y_{1}=2 y_{0}$. Therefore, $2 x_{0}$ is an integer. Let $n=2 x_{0}$. Then $y_{1}=2 y_{0}=2 k / x_{0}=4 k / n$. Thus, $n$ can take the value of any divisor of the number $4 k$. For $k=2020$, we need to find the number of integer divisors of the number $4 \cdot 2020=8080=2^{4} \cdot 5 \cdot 101$. The number of natural divisors of this number is $20=(4+1)(1+1)(1+1)$ (here we calculated the number of natural divisors using the powers of prime numbers in the prime factorization of the number 8080). Taking into account the negative divisors (corresponding to points of tangency in the third quadrant), we get double the number, i.e., a total of 40 points.