Difference between revisions of "2022 AMC 12A Problems/Problem 23"
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Therefore, the answer is <math>\boxed{\textbf{(D) 8}}</math>. | Therefore, the answer is <math>\boxed{\textbf{(D) 8}}</math>. | ||
− | NOTE: Detailed analysis of this problem (particularly the motivation and the proof of the lemma above) can be found in my video solution: | + | \bf NOTE: Detailed analysis of this problem (particularly the motivation and the proof of the lemma above) can be found in my video solution: |
https://youtu.be/4RHmsoDsU9E | https://youtu.be/4RHmsoDsU9E | ||
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) |
Revision as of 21:26, 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 .
\bf NOTE: Detailed analysis of this problem (particularly the motivation and the proof of the lemma above) can be found in my video solution:
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)