Question

In a machine, a particular gear completes a rotation in 6 seconds. Another gear completes a rotation in 9 seconds. At a certain instant, both gears complete their rotations at exactly the same time. How many seconds will elapse before the two gears complete their rotations at exactly the same time again?

Answer 1

36 seconds.

The first time the two gears will complete their rotations at the same time will be after 18 seconds. One gear will have rotated three times, the other, two. Keep counting the multiples of either number until you reach one that both 9 and 6 have. The next one after 18 is 36.