   SEARCH HOME Math Central Quandaries & Queries  Question from stefan, a student: how many years pass between the years when both 13 year and 17 year locusts are out at the same time? Stefan,

If they were 13 and 26 year locusts them they would both be out every 26 years since 13 × 2 = 26 and 26 × 1 = 26.

If they were 12 and 15 year locusts then they would both be out every 60 years since 12 × 5 = 60 and 15 × 4 = 60. I found 60 by writing the prime factorization of 12 and 15. 12 = 3 × 22 and 15 = 3 × 5. Hence if an integer is a multiple of 12 and also a multiple of 15 its prime factorization must contain two twos a three and a five. But 22 × 3 × 5 = 60 so 60 is the least number of years between the times both locusts are out.

Your numbers are 13 and 17 and both are prime numbers so any integer that is a multiple of 13 and 17 and hence its prime factorization must contain a thirteen and a seventeen. The smallest such integer is 13 × 17 = 221 and hence both locusts are out every 221 years.

I hope this helps,
Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.