9.4. Vasya cut out a triangle from cardboard and numbered its vertices with the digits 1, 2, and 3. It turned out that if Vasya's triangle is rotated 15 times clockwise around its vertex numbered 1 by the angle at this vertex, the triangle will return to its original position. If Vasya's triangle is rotated 6 times clockwise around its vertex numbered 2 by the angle at this vertex, the triangle will return to its original position. Vasya claims that if his triangle is rotated $n$ times around the vertex numbered 3 by the angle at this vertex, the triangle will return to its original position. What is the smallest $n$ that Vasya could have named so that his claim is true for at least some cardboard triangle?
Answer: 5.

Solution. Let the angle at the first vertex be $\alpha$ degrees, at the second $\beta$ degrees, and at the third $\gamma$ degrees. When rotating around the vertices, Vasya's triangle could make several full turns before returning to its starting position. Suppose that during the rotation around the first vertex it made $k$ full turns, around the second - $m$ full turns, and around the third - $l$ full turns. Then from the problem's condition, it follows that $15 \alpha = 360 k$, $6 \beta = 360 m$, and $n \gamma = 360 l$. From the first two equations, we find that $\alpha = 24 k$, $\beta = 60 m$, which means $\alpha \geq 24$, $\beta \geq 60$. Note that $n \geq 4$, otherwise $\gamma > 120$, which together with $\beta$ is more than $180$. $n \neq 4$, because otherwise $\gamma = 90$, so $\alpha + \beta = 90$, which is impossible given $\alpha = 24 k$, $\beta = 60 m$. $n = 5$ works: take a triangle with angles $\alpha = 48$, $\beta = 60$, $\gamma = 72$.

Comment. It is proven that $n \geq 4 - 3$ points.

It is proven that $n \neq 4 - 2$ points.

An example is provided for $n = 5 - 2$ points.

If the smallest value is not found, but an example of a triangle with such a property is provided -1 point.