13.301. In a certain mechanism, three gears of different diameters are connected in such a way that the largest one touches both of the smaller ones, and all three gears together have 60 teeth. When the largest gear is 20 teeth short of completing four full revolutions, the second and third gears make 5 and 10 full revolutions, respectively. How many teeth does each gear have individually?
## Solution.

Let $x, y, z$ be the number of teeth on three gears, where $x>y>z$ and $x+y+z=60$ (1). Over the time the gears rotate, the same number of teeth of each gear will come into contact, i.e., $10 z=5 y=4 x-20$ (2). Solving the system of equations (1),(2), we find $x=30, y=20, z=10$.

Answer: 10, 20, and 30 teeth.