Question

If $a,a+2$ and $a+4$ are prime numbers, then the number of possible solutions for $a$ is:
A. One
B. Two
C. Three
D. Four

Answer 1

Hint: First we recall the definition of prime numbers. Then we put the different values of $a$ in the given equations $a,a+2$ and $a+4$ to find the prime numbers. We have to find such a number that when we add 2 and 4 to the number, we will get prime numbers.

Complete step-by-step answer:
We know that a number is said to be a prime number if it can be divided by one and the number itself.
We have given that $a,a+2$ and $a+4$ are prime numbers.
We have to find the number of possible solutions for $a$.
We know that the smallest prime number is $2$.
So, when we take the value of $a=2$, we get
$a+2=2+2=4$, which is not a prime number. So, this is not the possible solution.
The next prime number is $3$.
When we take the value of $a=3$, we get
$a+2=3+2=5$ and $a+4=3+4=7$
$3,5,7$ are prime numbers.
Now, we check for $a=5$, we get
$a+2=5+2=7$ and $a+4=5+4=9$
As we know that $9$ is not a prime number. So, there is only one possible solution for $a$.
Option A is the correct answer.

Note: The number $1$ is not a prime number. A prime number has only two distinct factors, one and the number itself. $2$ is the only prime number which is even, all other prime numbers are odd numbers. The total number of prime numbers is infinite.