Stephan's grandfather told him about how cicadas sometimes damaged his orchard. One year, there were so many cicadas, they wrecked the buds on all of his young trees. Both the 13 -year and the 17 -year cicadas came up that year.B) 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? Explain.C) Stephan developed a method to determine the next time two types of cicadas will appear together. Ife says that if you multiply the cycles together, you get the next time that both types will appear together. Does Stephan's method work for any pair of cycles? If so, explain why. If not, provide an example for which Stephan's method does not work.

If 12 -year, 14 -year, and 16 -year cicadas that all appear this year, the next time they will all appear together can be determined by finding the lowest common multiple of 12, 14, and 16.

[tex]\begin{gathered} 12=2^2\times3 \\ 14=2\times7 \\ 16=2^4 \\ \text{Therefore:} \\ \text{LCM }=2^4\times3\times7 \\ =336 \end{gathered}[/tex]

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

Part C

Stephans method will work for the 13-year and 17-year cicadas since they are prime numbers.

However, it would not work in the case of the 12 -year, 14 -year, and 16 -year cicadas.

Going by Stephan's method, the next time the 12-year and 14-year cicadas will appear together will be:

[tex]=12\times14=168\text{ years}[/tex]

In actual fact, the next time they will appear together is the LCM of 12 and 14.

[tex]\begin{gathered} 12=2^2\times3 \\ 14=2\times7 \\ \text{LCM}=2^2\times3\times7=84\text{ years} \end{gathered}[/tex]