Question 36
Let $a$, $b$ and $n$ be positive integers. Denote by $K(n)$ the number of the representations of $1$ as a sum of $n$ numbers of the form $\frac{1}{k}$, where $k$ is a positive integer. Let $L(a, b)$ be the least positive integer $m$ such that the equation $\sum_{i=1}^{m} \frac{1}{x_{i}}=\frac{a}{b}$ has a solution in positive integers and set $L(b)=\max \{L(a, b), 1 \leq a \leq b\}$. Prove that the number of the positive divisors of $b$ does not exceed $2 L(b)+K(L(b)+2)$.