[ Greatest Common Divisor (GCD) and Least Common Multiple (LCM). Mutual simplicity ]

A wheel with a radius of 18 rolls around a circle with a radius of 40. A nail is driven into the wheel, which leaves marks on the circle each time it hits it. How many such marks will the nail leave on the circle? How many times will the wheel roll around the entire circle before the nail hits a point that has already been marked?

#
Let $l=4 \pi$. The length of the large circle is 201, and the small one is $9 l$. The nail will hit the previously marked point when the small circle rolls over an arc of length GCD$(20,9) l=180 l$, which means it will roll 9 times around the large circle. During this time, it will leave $180: 9=20$ marks.

## Answer

20 marks, 9 revolutions.

## Problem