Answer
See below
Work Step by Step
(a)
Six times will the two species emerge in the same year and compete to share the forest.
The life cycles of two species are 18 and 12 years, respectively.
The multiples of 18 less than or equal to 216 are \[18,36,54,72,90,108,126,144,162,180,198,216\].
The multiples of 12 less than or equal to 216 are \[12,24,36,48,60,72,84,96,108,120,132,144,156,168,180,192,204,216\].
From the two lists of numbers, the common multiples are \[36,72,108,144,180,216\].
The number of common multiples is 6.
Therefore, six times will the two species emerge in the same year and compete to share the forest.
(b)
The two species have to share the forest every 221 years.
The life cycles are 17 and 13, respectively, which are prime numbers.
The prime factor of \[17={{17}^{1}}\].
The prime factor of \[13={{13}^{1}}\].
Now, the least common multiple is \[17\times 13=221\].
Therefore, the two species have to share the forest every 221 years.
(c)
Part (a):
The life cycles of two species are 18 and 12 years, respectively.
The multiples of 18 less than or equal to 216 are \[18,36,54,72,90,108,126,144,162,180,198,216\].
The multiples of 12 less than or equal to 216 are \[12,24,36,48,60,72,84,96,108,120,132,144,156,168,180,192,204,216\].
From the two lists of numbers, the common multiples are \[36,72,108,144,180,216\].
The number of common multiples is 6.
Therefore, six times will the two species emerge in the same year and compete to share the forest.
Part (b):
The life cycles are 17 and 13, respectively, which are prime numbers.
The prime factor of \[17={{17}^{1}}\].
The prime factor of \[13={{13}^{1}}\].
Now, the least common multiple is \[17\times 13=221\].
Therefore, the two species have to share the forest every 221 years.
It is clear that in part (a) the species will share the forest every 36 years, but in part (b) they will share the forest every 221 years.
If they have prime number of years as the length of their life cycles,they will share the forest after a long time, that is,the product of the life cycle years.
