`P=441xx484xx529xx576xx625`. The total number of factors are

Options

A. 607
B. 5706
C. 1024
D. 6075

Answer

To find the total number of factors of the number \( P = 441 \times 484 \times 529 \times 576 \times 625 \), we will first break each number down into its prime factors, then use those factors to calculate the total number of factors.

### Step 1: Prime Factorization
Let's break down each number into its prime factors:

1. **441**:
\[ 441 = 21^2 = (3 \times 7)^2 = 3^2 \times 7^2 \]

2. **484**:
\[ 484 = 22^2 = (2 \times 11)^2 = 2^2 \times 11^2 \]

3. **529**:
\[ 529 = 23^2 \]

4. **576**:
\[ 576 = 24^2 = (2^3 \times 3)^2 = 2^6 \times 3^2 \]

5. **625**:
\[ 625 = 25^2 = (5^2)^2 = 5^4 \]

### Step 2: Combine the Prime Factors
Now, we will combine all the prime factors from each number:

\[ P = 441 \times 484 \times 529 \times 576 \times 625 \]

Combining the prime factors:
- From \( 441 \): \( 3^2 \times 7^2 \)
- From \( 484 \): \( 2^2 \times 11^2 \)
- From \( 529 \): \( 23^2 \)
- From \( 576 \): \( 2^6 \times 3^2 \)
- From \( 625 \): \( 5^4 \)

Now, we can sum the exponents of the same bases:
- For \( 2 \): \( 2 + 6 = 8 \) (from \( 484 \) and \( 576 \))
- For \( 3 \): \( 2 + 2 = 4 \) (from \( 441 \) and \( 576 \))
- For \( 5 \): \( 4 \) (from \( 625 \))
- For \( 7 \): \( 2 \) (from \( 441 \))
- For \( 11 \): \( 2 \) (from \( 484 \))
- For \( 23 \): \( 2 \) (from \( 529 \))

Thus, we have:
\[ P = 2^8 \times 3^4 \times 5^4 \times 7^2 \times 11^2 \times 23^2 \]

### Step 3: Calculate the Total Number of Factors
To find the total number of factors, we use the formula:
\[ \text{Total factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \]
where \( e_1, e_2, e_3, \ldots \) are the exponents of the prime factors.

Applying this to our prime factorization:
\[ \text{Total factors} = (8 + 1)(4 + 1)(4 + 1)(2 + 1)(2 + 1)(2 + 1) \]
\[ = 9 \times 5 \times 5 \times 3 \times 3 \times 3 \]

### Step 4: Calculate the Result
Now we will calculate:
\[ 9 \times 5 = 45 \]
\[ 45 \times 5 = 225 \]
\[ 225 \times 3 = 675 \]
\[ 675 \times 3 = 2025 \]
\[ 2025 \times 3 = 6075 \]

Thus, the total number of factors of \( P \) is **6075**.