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 $p_i$ .

Now, we use the result above to solve this problem.

Following from this lemma, the list of $n$ with $1 \leq n \leq 22$ and $k_n < L_n$ is $6, 7, 8, 18, 19, 20, 21, 22 .$

Therefore, the answer is $\boxed{\textbf{(D) 8}}$ .

$\textbf{NOTE: Detailed analysis of this problem}$ (particularly the motivation and the proof of the lemma above) $\textbf{can be found in my video solution:}$

https://youtu.be/4RHmsoDsU9E

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

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2022 AMC 12A Problems/Problem 23

Problem

Solution