HMMT February, 2019

Number theory Difficulty 5.0 AIME, harder Prove it United States

Problem:

How many ways are there to arrange the numbers $21, 22, 33, 35$ in a row such that any two adjacent numbers are relatively prime?

Solution

Solution:

$21$ cannot be adjacent to $33$ or $35$ , so it must be on one end bordering $22$ . $33$ cannot be adjacent to $21$ or $22$ , so it must be on the other end bordering $35$ . Thus, there are only $2$ orderings: $21, 22, 35, 33$ , and $33, 35, 22, 21$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.