Find all the integers that can be the measure in degrees of an interior angle of a regular polygon.
Solution. We know that the sum of the interior angles of an $n$-sided convex polygon is: $(n-2) \cdot 180^{\circ}$. If every interior angle of a regular polygon is equal, then the size of the interior angle at each vertex is:

$$
\frac{(n-2) \cdot 180^{\circ}}{n}=\frac{n \cdot 180^{\circ}-360^{\circ}}{n}=180^{\circ}-\frac{360^{\circ}}{n}
$$

We need to determine for which integer $n \geq 3$ the value of $180^{\circ}-\frac{360^{\circ}}{n}$ is also an integer. For this, we need to find the divisors of $360=2^{3} \cdot 3^{2} \cdot 5$, but we exclude the values 1 and 2 (polygons with such side numbers do not exist).

![](https://cdn.mathpix.com/cropped/2024_05_02_e1012dd0a3f345e4a199g-1.jpg?height=377&width=1162&top_left_y=527&top_left_x=468)

Following the arrows on the diagram, we multiply the prime powers. This way, we get each divisor of 360. The number of divisors is 24, and by excluding the two forbidden numbers (1 and 2), we get the possible values of $n$. For the 22 possibilities, we calculate the corresponding angles, which in degrees and in ascending order will be: $60,90,108,120,135$, $140,144,150,156,160,162,165,168,170,171,172,174,175,176,177,178,179$.