# Clusters on the Number Line

*English translation by SOTA – AI Community of the Russian original. Organisers who would like this translation removed can email sota.ai.community@gmail.com.*

*Task 1 of the school stage (Moscow) of the All-Russian School Olympiad (VsOSh) 2025/26 in artificial intelligence, grades 9–11. Original: [tasks-ai-9-11-sch-msk-25-26.pdf](https://vos.olimpiada.ru/upload/files/Arhive_tasks/2025-26/school/ai/tasks-ai-9-11-sch-msk-25-26.pdf).*

Let $k \geq 2$ be an integer. We say that two points on the number line are friends if the difference of their coordinates is divisible by $k$. A non-empty set of points is called a cluster if any two points in this set are friends and no point can be added to it so that this property is preserved.

For which integers $k \geq 2$ can the points of the number line with coordinates

$$
1,\ 7,\ 21,\ 22,\ 28,\ 42,\ 43,\ 49,\ 63
$$

be split into two clusters?

*Translator's note: the statement paper gives only the total for the olympiad (112 points); the official solutions give this task a maximum of 12 points (exact match of the answer).*