The number of 2 digit numbers having exactly 6 factors is :

Answer

To solve the problem of finding the number of two-digit numbers that have exactly 6 factors, we will follow these steps:

### Step 1: Understand the conditions for having exactly 6 factors
A natural number has exactly 6 factors if it can be expressed in one of the following two forms:
1. \( p^5 \) where \( p \) is a prime number.
2. \( p^2 \times q \) where \( p \) and \( q \) are distinct prime numbers.

### Step 2: Check for the first form \( p^5 \)
We will check if there are any two-digit numbers of the form \( p^5 \):
- The smallest prime number is 2.
- Calculate \( 2^5 = 32 \).
- The next prime is 3, and \( 3^5 = 243 \) which is not a two-digit number.

Thus, the only two-digit number of the form \( p^5 \) is **32**.

### Step 3: Check for the second form \( p^2 \times q \)
We will now check for numbers of the form \( p^2 \times q \):
- We will consider distinct primes \( p \) and \( q \).
- The possible values for \( p^2 \) must be less than 100.

1. **For \( p = 2 \)**:
- \( p^2 = 4 \)
- Possible \( q \) values (distinct primes): 3, 5, 7, 11, 13, 17, 19, 23.
- Calculate \( 4 \times q \):
- \( 4 \times 3 = 12 \)
- \( 4 \times 5 = 20 \)
- \( 4 \times 7 = 28 \)
- \( 4 \times 11 = 44 \)
- \( 4 \times 13 = 52 \)
- \( 4 \times 17 = 68 \)
- \( 4 \times 19 = 76 \)
- \( 4 \times 23 = 92 \)

The valid two-digit numbers are: **12, 20, 28, 44, 52, 68, 76, 92**.

2. **For \( p = 3 \)**:
- \( p^2 = 9 \)
- Possible \( q \) values (distinct primes): 2, 5, 7, 11, 13.
- Calculate \( 9 \times q \):
- \( 9 \times 2 = 18 \)
- \( 9 \times 5 = 45 \)
- \( 9 \times 7 = 63 \)
- \( 9 \times 11 = 99 \)

The valid two-digit numbers are: **18, 45, 63, 99**.

3. **For \( p = 5 \)**:
- \( p^2 = 25 \)
- Possible \( q \) values (distinct primes): 2, 3, 7, 11.
- Calculate \( 25 \times q \):
- \( 25 \times 2 = 50 \)
- \( 25 \times 3 = 75 \)

The valid two-digit numbers are: **50, 75**.

4. **For \( p = 7 \)**:
- \( p^2 = 49 \)
- Possible \( q \) values (distinct primes): 2, 3, 5, 11.
- Calculate \( 49 \times q \):
- \( 49 \times 2 = 98 \)

The valid two-digit number is: **98**.

### Step 4: List all the valid two-digit numbers
From our calculations, the two-digit numbers having exactly 6 factors are:
- From \( p^5 \): **32**
- From \( p^2 \times q \): **12, 20, 28, 44, 52, 68, 76, 92, 18, 45, 63, 99, 50, 75, 98**

### Step 5: Count the total numbers
Now, we count all the unique two-digit numbers:
- **12, 18, 20, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99**

Thus, the total count is **16**.

### Final Answer
The number of two-digit numbers having exactly 6 factors is **16**.

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