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?
In all gears, the number of teeth used during rotation is the same (each tooth is counted as many times as it has been in contact with another tooth on another gear). According to the problem, we can say the following about this number.

The first gear had 32 teeth and made four full turns and part of a fifth, so more than $32 \cdot 4=128$ and fewer than $32 \cdot 5=160$ teeth were used. The second gear had 24 teeth and made five full turns and part of a sixth, so more than $24 \cdot 5=120$ and fewer than $24 \cdot 6=144$ teeth were used. The third gear rotated exactly eight times, so the number of used teeth is divisible by eight.

In total, the number of used teeth is a number that is a multiple of eight, is greater than 128, and less than 144. The only such number is 136. The third gear had $136: 8=17$ teeth.

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