Exercise 8. Determine the number of natural numbers $n>2015$ such that $n$ is divisible by $n-2015$.
Solution to exercise 8 Let's write $n=2015+k$. We are looking for the number of positive integers $k$ such that $2015+k$ is divisible by $k$. This is equivalent to $k$ dividing $2015=5 \times 13 \times 31$. The divisors of 2015 are numbers of the form $5^{a} 13^{b} 31^{c}$ with $a, b, c$ equal to 0 or 1: there are 8 of them.