Understanding how many positive integers less than 100 have exactly 4 distinct positive factors requires a thorough examination of the number of factors a number can have based on its prime factorization. This knowledge is particularly useful in fields such as data science, computer science, and number theory, enhancing one's ability to work with number systems and algorithms.

Introduction

The number of divisors, or factors, of a positive integer can be determined from its prime factorization. For a number (n) expressed as

[n p_1^{e_1} times p_2^{e_2} times ldots times p_k^{e_k}]

where (p_i) are distinct prime numbers and (e_i) are their respective positive integer exponents, the total number of distinct positive factors (d(n)) is given by

[d(n) (e_1 1)(e_2 1) cdots (e_k 1)]

Determining the Count

We need to find how many positive integers less than 100 have exactly 4 distinct positive factors. This can be achieved by considering two primary cases.

Case 1:

When the number is of the form (n p^3), where (p) is a prime number. In this case, the number of factors is

[d(n) e_1 1 3 1 4]

So, we seek primes (p) such that (p^3 100).

Starting with the smallest primes:

(2^3 8) (3^3 27) (5^3 125) (too large)

The valid values here are (8) and (27).

Case 2:

When the number is of the form (n p_1 times p_2), where (p_1) and (p_2) are distinct prime numbers. Here, the number of factors is

[d(n) (1 1)(1 1) 2 times 2 4]

We need to find distinct primes (p_1) and (p_2) such that (p_1 times p_2 100).

The primes less than 100 are: (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97).

Now, we find valid pairs (p_1, p_2):

For (p_1 2): All pairs are valid, giving factors (6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94). For (p_1 3): All pairs are valid, giving factors (15, 21, 33, 39, 51, 57, 69, 87, 93). For (p_1 5): All pairs are valid, giving factors (35, 55, 65, 85). For (p_1 7): All pairs are valid, giving factors (77, 91).

For (p_1 11), the products exceed 100. Thus, the valid products are:

From (p_1 2): (6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94) (14 values) From (p_1 3): (15, 21, 33, 39, 51, 57, 69, 87, 93) (9 values) From (p_1 5): (35, 55, 65, 85) (4 values) From (p_1 7): (77, 91) (2 values)

Adding the valid numbers from both cases:

From (p^3): (8, 27) (2 values) From (p_1 times p_2): (6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94) (30 values)

Total count: (2 30 32)

Thus, the total number of positive integers less than 100 that have exactly 4 distinct positive factors is boxed{32}.