1. The picture shows several circles connected by segments. Nastya chooses a natural number \( n \) and places different natural numbers in the circles so that for all these numbers the following property holds:

if numbers \( a \) and \( b \) are not connected by a segment, then the difference \( a - b \) must be coprime with \( n \), and if they are connected, then \( a - b \) and \( n \) must have a common natural divisor greater than 1. For what smallest \( n \) does such an arrangement exist?

![](https://cdn.mathpix.com/cropped/2024_05_06_6bce8335c6b594326f89g-15.jpg?height=271&width=263&top_left_y=601&top_left_x=908)
Answer: $n=5 \cdot 7 \cdot 11=385$.

Solution. We will make two observations.

1) $n$ is not divisible by 2 and 3. Among seven numbers, there are always three numbers of the same parity. If $n$ is even, then they must be pairwise connected. Moreover, among seven numbers, there will be three numbers that give the same remainder when divided by 3. If $n$ is divisible by 3, then these numbers also form a three-link cycle. But such cycles do not exist in the picture.
2) If $p$ is a prime divisor of $n$, then among three consecutively connected numbers, there exists a pair of adjacent numbers whose difference is not divisible by $p$. Let $(a, b, c)$ be a chain of consecutively connected numbers where $a-b$ and $b-c$ are divisible by $p$. Then $a-c=(a-b)+(b-c) \vdots p$. This means that $(a, b, c)$ forms a three-link cycle, which is impossible.

We will show that the number $n$ has at least three distinct prime divisors. Let $p$ be a prime divisor of $n$. There are no more than three distinct pairs $(a, b)$ for which $a-b \vdots p$. If there were 4 such pairs, then some two of them would intersect, which contradicts 2). But in total, 7 pairs of numbers are connected by segments, so $n$ has at least three distinct prime divisors. In view of 1), the minimal possible values for them are $5, 7, 11$, that is, $n \geqslant 5 \cdot 7 \cdot 11=385$. The arrangement for $n=385$ is shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_6bce8335c6b594326f89g-15.jpg?height=320&width=340&top_left_y=1610&top_left_x=869)