2022 AMC 12A Problems/Problem 23
Problem
Let and
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 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.
20:20, 11 November 2022 (EST)20:20, 11 November 2022 (EST)LEMMA20:20, 11 November 2022 (EST)20:20, 11 November 2022 (EST)
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
.
20:20, 11 November 2022 (EST)20:20, 11 November 2022 (EST)20:20, 11 November 2022 (EST)20:20, 11 November 2022 (EST)20:20, 11 November 2022 (EST)
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 \boxed{\textbf{(D) 8}}.