A positive integer is called nice if it is a multiple of $8.$

Question

A certain nice positive integer $n$ has exactly $9$ positive divisors. How many prime numbers are divisors of $n?$

Answer 1

Let's first write $n$ as $2^a \cdot p_1^{b_1} \cdot p_2^{b_2} \cdots p_k^{b_k},$ where $p_1, p_2, \ldots, p_k$ are distinct prime numbers and $a, b_1, b_2, \ldots, b_k$ are positive integers.

Since $n$ has exactly $9$ positive divisors, we know that $n$ can be written in the form $p_1^2 \cdot p_2^2 \cdot 2^2$ or $p^8$ where $p$ is a prime number.

If $n = p_1^2 \cdot p_2^2 \cdot 2^2,$ then the number of prime divisors of $n$ is $2.$

If $n = p^8,$ then the number of prime divisors of $n$ is $1.$

Therefore, there are $\boxed{3}$ prime numbers that are divisors of $n,$ namely $2, p_1, p_2.$