There are four numbers whose product is 9261000 and each of these four numbers is formed by distinct prime numbers. The average of all the four numbers is :

Options

A. 61.75
B. 67.25
C. 82.33
D. data insufficient

Answer

To solve the problem, we need to find four distinct prime numbers whose product is 9261000, and then calculate the average of these four numbers.

### Step-by-Step Solution:

1. **Factor the Number 9261000**:
We start by factoring 9261000 into its prime factors.

- First, we can divide by 10 (which is 2 * 5):
\[ 9261000 \div 10 = 926100 \]
- Continuing to factor:
\[ 926100 \div 10 = 92610 \]
\[ 92610 \div 10 = 9261 \]
- Now we have:
\[ 9261000 = 10^3 \times 9261 = (2 \times 5)^3 \times 9261 \]

2. **Factor 9261**:
Next, we need to factor 9261. We can check for divisibility by small prime numbers:
- 9261 is odd, so it is not divisible by 2.
- The sum of the digits (9 + 2 + 6 + 1 = 18) is divisible by 3, so we divide by 3:
\[ 9261 \div 3 = 3087 \]
- Continuing with 3087, again the sum of the digits (3 + 0 + 8 + 7 = 18) is divisible by 3:
\[ 3087 \div 3 = 1029 \]
- Continuing with 1029:
\[ 1029 \div 3 = 343 \]
- Now, we factor 343, which is \(7^3\):
\[ 343 = 7 \times 7 \times 7 \]

3. **Combine the Factors**:
Now we can combine all the factors we found:
\[ 9261000 = 10^3 \times 9261 = (2 \times 5)^3 \times (3^3 \times 7^3) \]
- This gives us:
\[ 9261000 = 2^3 \times 5^3 \times 3^3 \times 7^3 \]

4. **Identify Distinct Prime Factors**:
The distinct prime factors are 2, 3, 5, and 7.

5. **Calculate the Average**:
The average of the four distinct prime numbers is calculated as follows:
\[ \text{Average} = \frac{\text{Sum of the numbers}}{\text{Number of numbers}} = \frac{2 + 3 + 5 + 7}{4} \]
- Calculate the sum:
\[ 2 + 3 + 5 + 7 = 17 \]
- Now calculate the average:
\[ \text{Average} = \frac{17}{4} = 4.25 \]

### Final Answer:
The average of the four numbers is **4.25**.