Who is asking: Parent
Level: Middle

Question:
JOHN CONJECTURED THAT BOTH 13-YR.&17-YR. LOCUSTS CAME OUT THE SAME YEAR. ASSUME IT IS CORRECT. HOW MANY YEARS PASS BETWEEN THE YRS. WHEN BOTH 13 YR. & 17YR. LOCUSTS ARE OUT AT THE SAME TIME? EXPLAIN. NEXT, SUPPOSE THERE WERE 12YR.,14 YR.,& 16YR. LO- CUSTS,& THEY ALL COME OUT THIS YR. HOW MANY YRS. WILL IT BE BEFORE THEY ALL COME OUT TOGETHER AGAIN? EXPLAIN

Hi,

Assume this year to be year 0 and that this year both the 13-yr and 17-yr locusts came out. You can make a table of values to keep track of how long it will take before they both come out together again, or you can reason it out as follows:

the 13-yr locusts will come out again in 13 years, 26 years, 39 years, 52 years, etc. (all multiples of 13)

the 17-yr locusts will come out again in 17 years, 34 years, 51 years, 68 years, etc. (all multiples of 17)

In order for them to come out together, the number of years must be a multiple of BOTH 13 AND 17. In other words, the first time they again appear together will coincide with the lowest (or least) common multiple of 13 and 17. This is the product of the two numbers, or 221 years from now. This is the earliest time when the number of years is divisible by both 13 and 17. It would then take another 221 years for this to happen again.

Can you see now how to do the second part of the problem?

Leeanne