There are two common types of periodical cicadas. The most prevalent is the 17 -year cicada, for which a brood of nymphs emerges from the ground to breed and lay eggs after a 17 -year underground wait. The 13 -year cicada has a similar life cycle, but with a 13 -year interval between one emergence and the next. Studies show that having a prime-number interval, such as 13 or 17 , offers evolutionary advantages to each species. In particular, there is less interference between the two species when they live in the same region. (a) If broods of 13 -and 17 -year species happened to emerge in the same year, how many years will pass before this interference is repeated? (b) Suppose, hypothetically, that there exist 12 - and 18 -year species of cicadas. If both species happened to emerge the same year, in how many years will they interfere with one another again? (a) Before the interference is repeated,

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We need to determine the least common multiple (LCM) for each pair of cicada life cycles.

### (a) 13-year and 17-year cicadas

1. Since 13 and 17 are both prime numbers, their LCM is simply their product:
   $$
   \text{LCM}(13, 17) = 13 \times 17 = 221
   $$
2. Thus, if the two species emerge in the same year, they will next interfere after $$221$$ years.

### (b) 12-year and 18-year cicadas

1. First, factorize the numbers:
   $$
   12 = 2^2 \times 3,\quad 18 = 2 \times 3^2
   $$
2. The LCM is found by taking the highest power of each prime factor:
   $$
   \text{LCM}(12, 18) = 2^2 \times 3^2 = 4 \times 9 = 36
   $$
3. Thus, for the hypothetical species, they would interfere again after $$36$$ years.

$$
\boxed{221 \text{ years for the 13- and 17-year cicadas, and } 36 \text{ years for the 12- and 18-year cicadas}}
$$
The 13- and 17-year cicadas will interfere again after 221 years. The 12- and 18-year cicadas will interfere again after 36 years.

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