The total number of prime factors of the product `(8)^(20), (15)^(24), (7)^(15)` is-

Options

A. 59
B. 98
C. 123
D. 138

Answer

To find the total number of prime factors of the product \( (8)^{20} \times (15)^{24} \times (7)^{15} \), we will follow these steps:

### Step 1: Prime Factorization of Each Term
First, we need to factor each base into its prime factors.

- **For \( 8 \)**:
\[ 8 = 2^3 \]
Therefore,
\[ (8)^{20} = (2^3)^{20} = 2^{60} \]

- **For \( 15 \)**:
\[ 15 = 3 \times 5 \]
Therefore,
\[ (15)^{24} = (3 \times 5)^{24} = 3^{24} \times 5^{24} \]

- **For \( 7 \)**:
\[ 7 \text{ is already a prime number.} \]
Therefore,
\[ (7)^{15} = 7^{15} \]

### Step 2: Combine the Prime Factors
Now we can combine all the prime factors from each term:
\[ (8)^{20} \times (15)^{24} \times (7)^{15} = 2^{60} \times 3^{24} \times 5^{24} \times 7^{15} \]

### Step 3: Count the Total Number of Prime Factors
The total number of prime factors is the sum of the exponents of the prime factors:
- The exponent of \( 2 \) is \( 60 \)
- The exponent of \( 3 \) is \( 24 \)
- The exponent of \( 5 \) is \( 24 \)
- The exponent of \( 7 \) is \( 15 \)

Now, we add these exponents together:
\[ 60 + 24 + 24 + 15 \]

### Step 4: Calculate the Sum
Calculating the sum:
\[ 60 + 24 = 84 \]
\[ 84 + 24 = 108 \]
\[ 108 + 15 = 123 \]

### Final Answer
Thus, the total number of prime factors of the product \( (8)^{20} \times (15)^{24} \times (7)^{15} \) is \( 123 \).

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