Question

If $a,a + 2,a + 4$ are prime numbers. Then the number of possible solution for $a.$ is___________
A) Three
B) Two
C) One
D) More than three

Answer 1

Hint- First of all we have to find a prime number so that if we add 2 to that number that number is a prime. Again if we add 4 with the number then the resulting number becomes prime.

Complete step by step answer:
We know that the smallest prime number is 2. 2 is an even number. So $a + 2$ and $a + 4$ are also even numbers. So they cannot be prime numbers. Only there is only one even positive prime number which is 2. The second prime number is 3. If $a = 3$ then $a + 2 = 3 + 2 = 5$ and $a + 4 = 3 + 4 = 7.$ clearly 5 and 7 are prime numbers. So when $a = 3$ we have $a,a + 2,a + 4$ are all prime numbers. There is no other prime number $a$ for which $a + 2,a + 4$ are also prime numbers.
So the number of positive solutions for $a$ is 1.
Hence, the correct answer is (C) one.

Additional information: A number p is said to be a prime number if it can be divided by only 1 and itself. There are two types of numbers. One is prime and the other is composite. The prime number starts from 2. The total number of prime numbers is infinite. Another definition says that a number p is said to be a prime number if it has only two distinct factors. According to this definition 1 is not a prime number because 1 has no two distinct factors. It has only one factor 1. For other prime numbers there are two distinct factors.

Note: There is only one even prime number which is 2. All other prime numbers are odd numbers. But the converse is not true. That is all the odd numbers are not necessarily prime numbers. The number 1 is neither prime nor composite.