A positive integer is called multiplicatively perfect if it is equal to the product of its proper divisors. For example, 10 is multiplicatively perfect since its proper divisors are 1,2 and 5 , and it is true that $1 \times 2 \times 5=10$. How many multiplicatively perfect integers are there between 2 and 30 ?
(A) 9
(B) 5
(C) 8
(D) 6
(E) 4
Suppose that $n$ is a multiplicatively perfect positive integer.

If $n$ can be written as $n=a \times b$ with $a$ and $b$ different positive integers, neither equal to 1 , then $n$ cannot have any other proper divisors other than 1 , since $n=1 \times a \times b$, so any other proper divisors would make this product too large.

So $n$ can only have 2 proper divisors other than 1 .

This tells us that $n$ cannot have more than two distinct prime divisors (otherwise, it would certainly have three proper divisors larger than one).

If $n$ has 2 distinct prime divisors, $p$ and $q$, then neither can occur in the prime factorization of $n$ more than once, otherwise $p^{2}$ or $q^{2}$ would be another proper divisor of $n$. So if $n$ has 2 distinct prime divisors, then $n=p q$ (which indeed has only three proper divisors: $1, p$ and $q$ ). The integers between 2 and 30 which are of the form $n=p q$ are $6,10,14,15,21,22$, and 26 , or a total of 7 integers.

If $n$ has only 1 prime factor, $p$, then $n=p^{3}$ in order to have exactly 3 proper divisors (namely, $1, p$ and $\left.p^{2}\right)$. Higher powers of $p$ will have more than 3 proper divisors and lower powers of $p$ will have fewer than 3 proper divisors. (For example, $p^{2}$ has 2 proper divisors and $p^{5}$ has 5 proper divisors.)

The integers between 2 and 30 which are of the form $n=p^{3}$ are 8 and 27 , or 2 integers in total. Therefore, there are 9 multiplicatively perfect numbers between 2 and 30 .

ANSWER: (A)