The total number of divisors of 10500 except 1 and it.

Options

A. 48
B. 50
C. 46
D. 56

Answer

To find the total number of divisors of 10500 except for 1 and itself, we can follow these steps:

### Step 1: Prime Factorization of 10500
First, we need to find the prime factorization of 10500.

1. **Divide by 2**:
- 10500 ÷ 2 = 5250
- 5250 ÷ 2 = 2625 (2 appears twice)

2. **Divide by 3**:
- 2625 ÷ 3 = 875 (3 appears once)

3. **Divide by 5**:
- 875 ÷ 5 = 175
- 175 ÷ 5 = 35
- 35 ÷ 5 = 7 (5 appears three times)

4. **Divide by 7**:
- 7 ÷ 7 = 1 (7 appears once)

Thus, the prime factorization of 10500 is:
\[ 10500 = 2^2 \times 3^1 \times 5^3 \times 7^1 \]

### Step 2: Calculate the Total Number of Divisors
To find the total number of divisors, we use the formula:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1)(e_4 + 1) \]
where \( e_1, e_2, e_3, e_4 \) are the powers of the prime factors.

From our factorization:
- For \( 2^2 \): \( e_1 = 2 \) → \( e_1 + 1 = 3 \)
- For \( 3^1 \): \( e_2 = 1 \) → \( e_2 + 1 = 2 \)
- For \( 5^3 \): \( e_3 = 3 \) → \( e_3 + 1 = 4 \)
- For \( 7^1 \): \( e_4 = 1 \) → \( e_4 + 1 = 2 \)

Now, we multiply these results:
\[ \text{Total Divisors} = (2 + 1)(1 + 1)(3 + 1)(1 + 1) = 3 \times 2 \times 4 \times 2 \]

Calculating this:
\[ 3 \times 2 = 6 \]
\[ 6 \times 4 = 24 \]
\[ 24 \times 2 = 48 \]

So, the total number of divisors of 10500 is 48.

### Step 3: Exclude 1 and 10500
Now, we need to exclude 1 and 10500 itself from the total count:
\[ \text{Divisors except 1 and 10500} = 48 - 2 = 46 \]

### Final Answer
The total number of divisors of 10500 except for 1 and itself is **46**.

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