13.314. Two gears are in mesh. Gear $A$ has 12 teeth, and gear $B$ has 54. How many revolutions will each gear make before both return to their initial position?
## Solution.

Let wheel $A$ make $x$ revolutions, and wheel $B$ make $y$ revolutions. Since the wheels are meshed, the same number of teeth of each wheel will come into contact over the time of their rotation, i.e., $12 x = 54 y$, from which $x = \frac{9}{2} y$. The number $x$ is a natural number, so the smallest value of $y = 2$.

Answer: 9 and 2.