'AB' is 2 digit prime number where A is not equal to B, if digits are reversed new number 'BA' is also a prime number. Then find sum of all such numbers.

Options

A. 374
B. 396
C. 407
D. 418

Answer

To solve the problem, we need to find all two-digit prime numbers 'AB' such that when the digits are reversed to form 'BA', the new number is also a prime number. We also need to ensure that the digits A and B are not equal. Finally, we will sum all such prime numbers.

### Step-by-step Solution:

1. **Identify Two-Digit Prime Numbers**:
The two-digit prime numbers are:
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

2. **Check Each Prime Number**:
For each prime number, we will reverse the digits and check if the resulting number is also a prime number.

3. **List of Valid Prime Numbers**:
- **13**: Reversed is **31** (both prime)
- **17**: Reversed is **71** (both prime)
- **31**: Reversed is **13** (both prime)
- **37**: Reversed is **73** (both prime)
- **71**: Reversed is **17** (both prime)
- **73**: Reversed is **37** (both prime)
- **79**: Reversed is **97** (both prime)
- **97**: Reversed is **79** (both prime)

4. **Sum of Valid Numbers**:
The valid prime numbers that meet the criteria are:
- 13, 17, 31, 37, 71, 73, 79, 97.

Now, we will sum these numbers:
\[ 13 + 17 + 31 + 37 + 71 + 73 + 79 + 97 = 418. \]

5. **Final Answer**:
The sum of all such two-digit prime numbers is **418**.