The average of 3 prime numbers lying between 47 and 74 is 191/3.The greatest possible diffrence between any two out of the 3 prime number is:

Options

A. 12
B. 14
C. 18
D. can't be determined

Answer

To solve the problem, we need to find three prime numbers between 47 and 74 such that their average is \( \frac{191}{3} \). Then, we will determine the greatest possible difference between any two of these prime numbers.

### Step 1: Identify the prime numbers between 47 and 74.
The prime numbers in this range are:
- 53
- 59
- 61
- 67
- 71
- 73

### Step 2: Calculate the total sum of the three prime numbers.
Given that the average of the three prime numbers is \( \frac{191}{3} \), we can find the total sum of these numbers by multiplying the average by 3:
\[ \text{Total Sum} = \frac{191}{3} \times 3 = 191 \]

### Step 3: Find combinations of three prime numbers that sum to 191.
We need to find three prime numbers from our list that add up to 191. We can check combinations of the prime numbers:

1. **Combination 1:** \( 53 + 67 + 71 = 191 \)
2. **Combination 2:** \( 59 + 61 + 71 = 191 \)

Both combinations work, so we can use either set of prime numbers.

### Step 4: Calculate the differences between the prime numbers.
Next, we will calculate the differences between the numbers in each combination to find the greatest possible difference.

#### For Combination 1: \( 53, 67, 71 \)
- Difference between 71 and 53:
\[ 71 - 53 = 18 \]
- Difference between 71 and 67:
\[ 71 - 67 = 4 \]
- Difference between 67 and 53:
\[ 67 - 53 = 14 \]

The greatest difference in this combination is **18**.

#### For Combination 2: \( 59, 61, 71 \)
- Difference between 71 and 59:
\[ 71 - 59 = 12 \]
- Difference between 71 and 61:
\[ 71 - 61 = 10 \]
- Difference between 61 and 59:
\[ 61 - 59 = 2 \]

The greatest difference in this combination is **12**.

### Step 5: Determine the greatest possible difference.
Comparing the greatest differences from both combinations, the maximum difference is:
\[ \text{Greatest Difference} = 18 \]

### Final Answer:
The greatest possible difference between any two of the three prime numbers is **18**.

---