Is $\sqrt[3]{24+\sqrt[3]{24+\sqrt[3]{24+\cdots}}}$ even? prime? composite? an even prime?

Question

I am not being able to solve this question. It is basically a high school level question and chapter I think is basic fundamentals of maths.

The value of $$\sqrt[3]{24+\sqrt[3]{24+\sqrt[3]{24+\cdots}}}$$ is equal to ...

1. an even number
2. a prime number
3. a composite number
4. an even prime number

Answer 1

Let us call this number X. Cubing it we have

$X^3=X+24$ so $X$ is the solution of $X^3-X-24=0$ substituting( or using Wolfram) the divisors of $24$ we have that $3$ is the only real solution.