Question

Suppose there are 12-year, 14-year, and 16-year cicadas that all appear this year. After how many years will all three types of cicadas appear together again? How do you know?

Answer 1

The 12-year, 14-year, and 16-year cicadas will appear together again in 336 years, as 336 is the least common multiple (LCM) of the three numbers, found by listing the multiples of each until a common multiple is identified.

Step-by-step explanation:

To find out after how many years the 12-year, 14-year, and 16-year cicadas will all appear together again, we need to calculate the least common multiple (LCM) of the three numbers. The LCM of a set of numbers is the smallest number that is a multiple of each number in the set.

Calculating the LCM

1. List the multiples of each number until we find a common multiple.
2. 12-year cicadas: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ..., 336.
3. 14-year cicadas: 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, ..., 336.
4. 16-year cicadas: 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, ..., 336.
5. Note that the number 336 appears in all three lists, so it is the LCM of 12, 14, and 16.

Therefore, all three types of cicadas will appear together again in 336 years.