how may 2 digit numbers have exactly 4 positive factors(here 1 and n are also considered factors of n)

To find how many two-digit numbers have exactly 4 positive factors, we can approach the problem by examining the structure of numbers and their divisors.

### Key Concept:
A number \( n \) has exactly 4 positive factors if and only if it is in one of the following forms:
1. \( n = p^3 \), where \( p \) is a prime number.
2. \( n = p \times q \), where \( p \) and \( q \) are distinct prime numbers.

### Case 1: \( n = p^3 \)
For \( n \) to be a two-digit number:
- \( p^3 \) must be between 10 and 99.

Let's check which primes satisfy this:
- \( 2^3 = 8 \) (Not a two-digit number)
- \( 3^3 = 27 \)
- \( 5^3 = 125 \) (Exceeds two digits)

So, only \( n = 27 \) works in this form.

### Case 2: \( n = p \times q \)
For \( n \) to be a two-digit number:
- \( p \times q \) must be between 10 and 99.

Let's check combinations of distinct primes:
- \( 2 \times 3 = 6 \) (Not a two-digit number)
- \( 2 \times 5 = 10 \)
- \( 2 \times 7 = 14 \)
- \( 2 \times 11 = 22 \)
- \( 2 \times 13 = 26 \)
- \( 2 \times 17 = 34 \)
- \( 2 \times 19 = 38 \)
- \( 2 \times 23 = 46 \)
- \( 2 \times 29 = 58 \)
- \( 2 \times 31 = 62 \)
- \( 2 \times 37 = 74 \)
- \( 2 \times 41 = 82 \)
- \( 2 \times 43 = 86 \)
- \( 2 \times 47 = 94 \)
- \( 2 \times 53 = 106 \) (Exceeds two digits)
- \( 3 \times 5 = 15 \)
- \( 3 \times 7 = 21 \)
- \( 3 \times 11 = 33 \)
- \( 3 \times 13 = 39 \)
- \( 3 \times 17 = 51 \)
- \( 3 \times 19 = 57 \)
- \( 3 \times 23 = 69 \)
- \( 3 \times 29 = 87 \)
- \( 3 \times 31 = 93 \)
- \( 5 \times 7 = 35 \)
- \( 5 \times 11 = 55 \)
- \( 5 \times 13 = 65 \)
- \( 5 \times 17 = 85 \)
- \( 5 \times 19 = 95 \)
- \( 7 \times 11 = 77 \)
- \( 7 \times 13 = 91 \)

### Counting the Results:
- From Case 1: \( n = 27 \) (1 number)
- From Case 2: There are 29 valid two-digit products listed above.

Adding these cases together, we get:
\[
1 \text{ (from Case 1)} + 29 \text{ (from Case 2)} = 30 \text{ numbers}
\]

Thus, there are **30 two-digit numbers** that have exactly 4 positive factors.

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Would you like more details on the calculation process or further explanation? Here are some related questions you might be interested in:

1. What are the factors of a number?
2. How do you determine the number of factors a number has?
3. What are prime numbers and how are they relevant to factors?
4. Can a number have exactly 6 positive factors? 
5. How many two-digit numbers have exactly 3 positive factors?
6. What is the relationship between the prime factorization of a number and its total number of factors?
7. Are there any two-digit numbers with exactly 5 positive factors?
8. How do you calculate the total number of factors using prime factorization?

**Tip:** Understanding the structure of a number's prime factorization is key to determining its number of factors.

Discover how many two-digit numbers have exactly 4 positive factors through detailed mathematical analysis. Explore prime factorization and number theory concepts to solve this problem. Suitable for students in Grades 7-9.

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