## 23. Polygons

Divide a circle into $n$ equal arcs. Then connect the resulting points by $p$ [i.e., the 1st with the ($p+1$)th, the 2nd with the ($p+2$)th, and so on, until you return to the starting point. How many sides will the resulting polygon have?

Remark. It is useful to check the general formula obtained for $n=10$ and $p=2,3$, and 4.
23. If $p$ is a divisor of $n$, then the number of sides of the polygon is $n / p$. If $p$ is not a divisor of $n$, then the fraction $n / p$ should be multiplied by $x$ - the number of corresponding circumferences: $x=\inf (y$ such that $(y n / p)$ is an integer). In other words,

$$
\text { number of sides }=\frac{n x}{p}=\frac{\operatorname{LCM}(n, p)}{p},
$$

where $\operatorname{LCM}(n, p)$ is the least common multiple of $n$ and $p$.

![](https://cdn.mathpix.com/cropped/2024_05_21_fe999c0fe2ad81fc5164g-164.jpg?height=634&width=606&top_left_y=1322&top_left_x=725)

If $n=10$ and $p=2$, then the number of sides is $10 / 2=5$. If $n=10$ and $p=3$, then the number of sides is $30 / 3=10$. If $n=10$ and $p=4$, then the number of sides is $20 / 4=5$.