The number of divisors a number 38808 can have, excluding 1 and the number itself is:

Options

A. 70
B. 72
C. 71
D. None

Answer

To find the number of divisors of the number 38808, excluding 1 and the number itself, we will follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of 38808.

1. **Divide by 2** (the smallest prime number):
- 38808 ÷ 2 = 19404
- 19404 ÷ 2 = 9702
- 9702 ÷ 2 = 4851 (stop here as 4851 is odd)

2. **Divide by 3** (next smallest prime):
- 4851 ÷ 3 = 1617
- 1617 ÷ 3 = 539
- 539 is not divisible by 3.

3. **Divide by 7**:
- 539 ÷ 7 = 77
- 77 ÷ 7 = 11
- 11 is a prime number.

So, the prime factorization of 38808 is:
\[ 38808 = 2^3 \times 3^2 \times 7^2 \times 11^1 \]

### Step 2: Calculate the Total Number of Divisors
The formula to calculate the total number of divisors (D) from the prime factorization is:
\[ D = (e_1 + 1)(e_2 + 1)(e_3 + 1)(e_4 + 1) \]
where \( e_1, e_2, e_3, e_4 \) are the powers of the prime factors.

For our factorization:
- \( e_1 = 3 \) (for 2)
- \( e_2 = 2 \) (for 3)
- \( e_3 = 2 \) (for 7)
- \( e_4 = 1 \) (for 11)

Plugging in the values:
\[ D = (3 + 1)(2 + 1)(2 + 1)(1 + 1) \]
\[ D = 4 \times 3 \times 3 \times 2 \]

### Step 3: Calculate the Product
Now, we will calculate the product:
- First, calculate \( 4 \times 3 = 12 \)
- Next, calculate \( 12 \times 3 = 36 \)
- Finally, calculate \( 36 \times 2 = 72 \)

So, the total number of divisors of 38808 is 72.

### Step 4: Exclude 1 and the Number Itself
Since we need to exclude 1 and the number itself from the total divisors:
\[ \text{Divisors excluding 1 and 38808} = 72 - 2 = 70 \]

### Final Answer
Thus, the number of divisors of 38808, excluding 1 and the number itself, is **70**.

---