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. What is the smallest possible value of $n?$

Answer 1

To have exactly $9$ positive divisors, $n$ must be in the form $p^2$ or $p \cdot q$ where $p$ and $q$ are distinct primes. Since $n$ is a multiple of $8$, it must be $8$, $2^3 \cdot p$, or $2^3 \cdot p^2$.

If $n = 8$, then $8$ only has $4$ positive divisors $(1, 2, 4, 8)$.
If $n = 2^3 \cdot p$ for some prime $p$, then $n$ has $(3+1)(1+1) = 8$ positive divisors.
If $n = 2^3 \cdot p^2$ for some prime $p$, then $n$ has $(3+1)(2+1) = 12$ positive divisors.

Thus, the smallest possible value of $n$ is $2^3 \cdot p = 8 \cdot 3 = \boxed{24}.$