2022 AMC 12A Problems/Problem 23
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 .
textbf{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)