3-2. Two identical polygons were cut out of cardboard, aligned, and pierced with a pin at some point. When one of the polygons is rotated around this "axis" by $25^{\circ} 30^{\prime}$, it aligns again with the second polygon. What is the smallest possible number of sides of such polygons?
Solution 2. Answer: 240. First, note that $\frac{1}{360} \cdot 25 \frac{1}{2}=\frac{17}{240}$, and the numbers 17 and 240 are coprime. Consider a ray emanating from the "axis" to the vertex of the first polygon. Rotations of this ray around the "axis" by angles $k \cdot 25^{\circ} 30^{\prime}$, where $k=1,2, \ldots, 240$, are distinct. Indeed, if the rotations of the ray by angles $k_{1} \cdot 25^{\circ} 30^{\prime}$ and $k_{2} \cdot 25^{\circ} 30^{\prime}$ coincide, then the number $\frac{\left(k_{1}-k_{2}\right) 17}{240}$ is an integer, which means that $k_{1}-k_{2}$ is divisible by 240. On each of these 240 rays, there is a vertex of the polygon, so the number of sides of the polygon is at least 240. On the other hand, when a regular 240-gon is rotated by an angle of $25^{\circ} 30^{\prime}$ around its center, it coincides with itself.