2022 AMC 12A Problems/Problem 23

Problem

Let $h_n$ and $k_n$ be the unique relatively prime positive integers such that

$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \frac{h_n}{k_n} .$

Let $L_n$ denote the least common multiple of the numbers $1,2,3,\cdots,n$ . For how many integers $n$ with $1 \leq n \leq 22$ is $k_n < L_n$ ?

Solution

We will use the following lemma to solve this problem.

Denote by $p_1^{\alpha_1} p_2^{\alpha_2} \cdots p_m^{\alpha_m}$ the prime factorization of $L_n$ . For any $i \in \left\{ 1, 2, \cdots, m \right\}$ , denote $\sum_{j = 1}^{\left\lfloor \frac{n}{p_i^{\alpha_i}} \right\rfloor} \frac{1}{j} = \frac{a_i}{b_i}$ , where $a_i$ and $b_i$ are relatively prime. Then $k_n = L_n$ if and only if for any $i \in \left\{ 1, 2, \cdots, m \right\}$ , $a_i$ is not a multiple of $p_i$ .