Why don't 17 year old cicadas and 13 year old cicadas appear together?

This question is equivalent to asking what multiples of 13 and 17 are equal, and more specifically, what is the lowest multiple of 13 that equals a multiple of 17 or what is the lowest multiple of 17 that equals a multiple of 13. Since 13 and 17 are prime, they have no factors in common. This means that the lowest common multiple is equal to 13 x 17 = 221. Therefore, it will be 221 years before the 13 and 17 year cicadas appear together again.