How many numbers between 1 and 100, inclusive, have exactly 5 positive

We should recall the rule we can use to determine the total number of factors:

For a positive integer n (where n > 1),
i) if the prime factorization of n is p^a (where p is a prime), then the total number of factors of n is equal to a + 1.
ii) if the prime factorization of n is p^a * q^b (where p and q are distinct primes), then the total number of factors of n is equal to (a + 1)(b + 1).
(Note: The concept can be extended to the prime factorization of n when n has 3 or more distinct prime factors.)

Since 5 is a prime number, the number(s) we seek can’t have more than 1 prime factor. For example, if it has two distinct prime factors, then, according to the rule, the total number of factors is equal to (a + 1)(b + 1). But (a + 1)(b + 1) can’t be equal to 5 since 5 is a prime number..

Thus we see that the number must have only 1 prime factor and is of the form p^a. Since, according to the rule, the total number of factors is equal to a + 1, we can set a + 1 = 5 and obtain a = 4. Now let’s check some values of p (keep in mind that p is a prime):

If p = 2, then p^4 = 2^4 = 16 (which is between 1 and 100)
If p = 3, then p^4 = 3^4 = 81 (which is between 1 and 100)
If p = 5, then p^4 = 5^4 = 625 (which is more than 100)

Thus, there are only two integers between 1 and 100 (6 and 81) that have exactly 5 positive factors.

Answer: C