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, 2021

Combinatorics Difficulty 5.1 AIME, harder Prove it United States

Problem:
Find the number of ways in which the nine numbers
$1, 12, 123, 1234, \ldots, 123456789$
can be arranged in a row so that adjacent numbers are relatively prime.

Solution

Solution:
The six numbers $12, 123, 12345, 123456, 12345678$ , and $123456789$ are divisible by $3$ , so they cannot be adjacent. However, arranging six numbers in a row with no two adjacent requires at least $11$ numbers, which is impossible.

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