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