+318 A positive integer is called nice if it is a multiple of $8.$
A certain nice positive integer $n$ has exactly $9$ positive divisors. What is the smallest possible value of $n?$
+135 There are 2 cases:
\(n = p_1 ^ 8\) or \(n = p_1 ^ 2 p_2 ^ 2\)
But wait! \(8 = 2^3\), so the second possibility won't work since it must not be divisible by a power of 2 greater than 4. Therefore, the smallest (and only) ice number with exactly 9 divisors is \(2^8 = \mathbf{256}\)
