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 | ||
− | + | <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} . | ||
− | + | </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 and
be the unique relatively prime positive integers such that
Let denote the least common multiple of the numbers
. For how many integers
with
is
?
Solution
We will use the following lemma to solve this problem.
Denote by the prime factorization of
.
For any
, denote
, where
and
are relatively prime.
Then
if and only if for any
,
is not a multiple of
.
Now, we use the result above to solve this problem.
Following from this lemma, the list of with
and
is
\[
6, 7, 8, 18, 19, 20, 21, 22 .
\]
Therefore, the answer is .