The proper divisors of a positive integer, $n$, are all the divisors excluding $n$ itself. For example, the proper divisors of $6$ are $1, 2,$ and $3$.
A number, $n$, is said to be multiplicatively perfect if the product of its proper divisors equals $n$. The smallest such example is six: $6 = 1 \times 2 \times 3$ the next such example is eight: $8 = 1 \times 2 \times 4$.
Determine the nature of all multiplicatively perfect numbers.