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AMC (AIMO validation set) / https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_23 / Let h_(n) and k_(n) be the unique relatively prime positive integers such that (1)/(1)+(1)/(2)+(1)/(3)+⋯+(1)/(n)=(h_(n))/(k_(n)). Let L_(n) denote the least common multiple of the …

Problem

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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, \ldots, n

. For how many integers with

1\le{n}\le{22}

k_n<L_n

Plain-text mathematical notation (without MathML)

Let h_(n) and k_(n) be the unique relatively prime positive integers such that (1)/(1)+(1)/(2)+(1)/(3)+⋯+(1)/(n)=(h_(n))/(k_(n)). Let L_(n) denote the least common multiple of the numbers 1,2,3,…,n. For how many integers with 1≤n≤22 is k_(n)<L_(n)?

Original LaTeX notation

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 $1, 2, 3, \ldots, n$. For how many integers with $1\le{n}\le{22}$ is $k_n<L_n$?

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Source and history

upstream_id / Original problem ID: https://artofproblemsolving.com/wiki/index.php/2022_AMC_12A_Problems/Problem_23
task_id: 85031697-9ecf-5312-ba59-8470c023517e
task_key: default--train--https~3a~2f~2fartofproblemsolving~2ecom~2fwiki~2findex~2ephp~2f2022~5fAMC~5f12A~5fProblems~2fProblem~5f23
task_revision_id: 2
config / split: default / train

Official source

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