When Will All Three Cicadas Emerge Again?

Question:

If there were a 12-year cicada, a 15-year cicada, and a 10-year cicada present in the same year, after how many years will all three cicadas come out of the ground again?

Answer:

The least common multiple (LCM) of the life cycles of 12-year, 15-year, and 10-year cicadas is 60 years.

To determine when all three types of cicadas will emerge out of the ground again together, we need to find the least common multiple (LCM) of their life cycles. The LCM is the smallest number that is a multiple of each of the given numbers. In this case, the life cycles are 12 years, 15 years, and 10 years.

First, let's list the multiples of each number:

- 12-year cicada: 12, 24, 36, 48, 60, 72, ...

- 15-year cicada: 15, 30, 45, 60, 75, ...

- 10-year cicada: 10, 20, 30, 40, 50, 60, ...

Since 60 is the smallest number that appears in all three lists, the LCM of 12, 15, and 10 is 60 years. After 60 years, all three types of cicadas will emerge out of the ground again together.