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
We start with the number 38808 and perform prime factorization.

- Divide by 2:
\[ 38808 \div 2 = 19404 \quad (2^1) \]
- Divide by 2 again:
\[ 19404 \div 2 = 9702 \quad (2^2) \]
- Divide by 2 again:
\[ 9702 \div 2 = 4851 \quad (2^3) \]
- Now, 4851 is not divisible by 2. Let's try dividing by 3:
\[ 4851 \div 3 = 1617 \quad (3^1) \]
- Divide by 3 again:
\[ 1617 \div 3 = 539 \quad (3^2) \]
- Now, 539 is not divisible by 3 or 5. Let's try dividing by 7:
\[ 539 \div 7 = 77 \quad (7^1) \]
- Finally, factor 77:
\[ 77 = 7 \times 11 \quad (7^2 \text{ and } 11^1) \]

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

### Step 2: Calculate the Number of Divisors
The formula for finding the number of divisors from the prime factorization \( p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_n^{e_n} \) is given by:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots (e_n + 1) \]

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

Now, applying the formula:
\[ (3 + 1)(2 + 1)(2 + 1)(1 + 1) = 4 \times 3 \times 3 \times 2 \]

### Step 3: Calculate the Product
Now we calculate:
\[ 4 \times 3 = 12 \]
\[ 12 \times 3 = 36 \]
\[ 36 \times 2 = 72 \]

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 (38808), we subtract 2 from the total number of divisors:
\[ 72 - 2 = 70 \]

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

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