Question

A three-digit prime number is such that the digit in the unit place is equal to the sum of the other two, and if the other digits are interchanged, we still have a prime number of three digits. Then the total number of such primes is:
 $ (a){\text{ 2}} $
 $ (b){\text{ 4}} $
 $ (c){\text{ 6}} $
 $ (d){\text{ 3}} $

Answer 1

Hint: In the above given question, we must know that the prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. Here, we will check the given conditions for all the 1- digit odd numbers one by one to find out the required prime numbers.

Complete step-by-step answer:
We are given that the three-digit number is a prime number, which implies that the last digit of the three-digit number is odd.
Now, we will check for all the possibilities 1,3,5,7,9,
Here, the last digit 1 is not possible because it will not satisfy the given condition of summing of the other two numbers.
For 3, the other 2 digits would be 1 and 2. The 3- digit numbers so obtained would be either 123 or 213, none of these is a prime number.
For 5, the numbers ending with 5 are not the prime numbers.
Similarly, for 9, the digit sum would be 1,8 and 2,7 and 3,6 and 4,5 and we know that, none of these is a prime number and thus rejected.
Now, for 7, the digit sum can be 1,6 and 2,5 and 3,4.
Thus, the possible numbers can be 167,617,347,437,257,527
Where 527 is divisible by 17 and 437 is divisible by 19.
Therefore, the total prime numbers are 167,617,347,257.
Hence, there are 4 primes.
So, the correct solution is the option $ (b) $ .

Note: Check the given conditions for all the odd numbers individually. The unit digit number can satisfy the condition of being prime but the other conditions are also needed to be fulfilled. Make sure you check all the possible combinations carefully.