10. The dividing circle of the dividing head is uniformly marked with 24 holes. What regular polygons can be marked using this instrument?
10. By connecting all 24 holes sequentially, we get a twenty-four-sided polygon, connecting every other hole - a twelve-sided polygon,

![](https://cdn.mathpix.com/cropped/2024_05_21_ec9ddf54a7c86775293eg-076.jpg?height=182&width=528&top_left_y=1985&top_left_x=1181)
connecting every two holes - an eight-sided polygon, connecting every three holes - a six-sided polygon, and connecting every seven holes - a three-sided polygon. If $n$ is the number of vertices of the inscribed polygon, $3 \leqslant n \leqslant 24$, then $\frac{24-n}{n}=k-$ $5^{*}$

![](https://cdn.mathpix.com/cropped/2024_05_21_ec9ddf54a7c86775293eg-077.jpg?height=183&width=577&top_left_y=248&top_left_x=357)

is an integer. Then $n=\frac{24}{k+1}$. Therefore, $0 \leqslant k \leqslant 7$.

Answer. Using a dividing circle with 24 holes, 6 different regular polygons can be marked.