Bětka was playing with gears, which she arranged as indicated in the picture. When she then turned one gear, all the others started turning as well. In the end, she was satisfied with a gear system where the first gear had 32 and the second 24 teeth. When the third gear made exactly eight turns, the second gear made five turns and part of a sixth, and the first gear made four turns and part of a fifth.

Determine how many teeth the third gear had.

(E. Novotná)

![](https://cdn.mathpix.com/cropped/2024_04_17_848c3145926e37c65b12g-3.jpg?height=434&width=837&top_left_y=1082&top_left_x=612)

Hint. How many times will the teeth of the first gear engage with the teeth of the second gear if the first gear turns four times?