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Difference between revisions of "2022 AMC 12A Problems/Problem 23"

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Let <math>h_n</math> and <math>k_n</math> be the unique relatively prime positive integers such that
 
Let <math>h_n</math> and <math>k_n</math> be the unique relatively prime positive integers such that
  
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\[
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<cmath>
 
\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \frac{h_n}{k_n} .
 
\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \frac{h_n}{k_n} .
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\]
+
</cmath>
  
 
Let <math>L_n</math> denote the least common multiple of the numbers <math>1,2,3,\cdots,n</math>. For how many integers <math>n</math> with <math>1 \leq n \leq 22</math> is <math>k_n < L_n</math>?
 
Let <math>L_n</math> denote the least common multiple of the numbers <math>1,2,3,\cdots,n</math>. For how many integers <math>n</math> with <math>1 \leq n \leq 22</math> is <math>k_n < L_n</math>?

Revision as of 20:22, 11 November 2022

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 $\boxed{\textbf{(D) 8}}$.

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