Difference between revisions of "2022 AMC 12A Problems/Problem 23"
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==Solution 1== | ==Solution 1== | ||
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+ | We are given that <cmath>\sum_{i=1}^{n}\frac1i = \frac{1}{L_n}\sum_{i=1}^{n}\frac{L_n}{i} = \frac{h_n}{k_n}.</cmath> Since <math>k_n < L_n,</math> we need <math>\gcd\left(\sum_{i=1}^{n}\frac{L_n}{i}, L_n\right)>1.</math> | ||
==Solution 2== | ==Solution 2== |
Revision as of 22:50, 3 January 2023
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 1
We are given that Since
we need
Solution 2
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
Therefore, the answer is .
Note: Detailed analysis of this problem (particularly the motivation and the proof of the lemma above) can be found in my video solution below.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~MathProblemSolvingSkills.com
See Also
2022 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.