4. The teacher gave Vasya and Petya two identical cardboard $n$-gons. Vasya cut his polygon into 33-gons along non-intersecting diagonals, while Petya cut his polygon into 67-gons along non-intersecting diagonals. Find the smallest possible value of $n$.
Answer: 2017.

The sum of the angles of an $n$-sided polygon is $(n-2) \cdot 180^{\circ}$. If it is cut into $k$ 33-sided polygons, then $(n-2) \cdot 180^{\circ}=k \cdot (33-2) \cdot 180^{\circ}$, hence $n-2 \vdots 31$. Similarly, from the second condition, it follows that $n-2 \vdots 65$. Since 31 and 65 are coprime, the smallest possible value of $n=31 \cdot 65+2=$ 2017.

$\pm$ Correct solution with a minor error leading to an incorrect answer. 5 points

- The solution includes the idea of calculating the sum of angles, but lacks conclusions about divisibility. 2 points
- The correct answer is given, but not justified. In particular, a certain pair of cuts for which "everything fits" is provided. 1 point