Question

State true (T) or false (F).
The sum of primes cannot be a prime
\[\begin{align}
  & \text{(A) True} \\
 & \text{(B) False} \\
\end{align}\]

Answer 1

Hint: First of all, we need to know the definition of prime number. A prime number is a number which does not have factors other than 1 or itself. We are having two types of prime numbers. The first type is called as odd prime numbers and the other is called as even prime numbers. We know that 2 is the only even prime number. To verify that the sum of primes cannot be a prime, let us consider 3 cases. In the first case, we take odd primes and we will check the results. In the second case we will take one as an odd prime and another as 2 and then we will check the results. Finally, in the third case we will check if the sum of 2 and 2 is prime or not. By considering the results of three cases, we make a final statement in the form of a statement.

Complete step by step solution:
Before solving the question, we should know the definition of a prime number. A prime number is a number which does not have factors other than 1 or itself. We know that 2 is the only even prime number. So, the remaining prime numbers other 2 are odd prime numbers.
Case -1:
Let us assume any two prime numbers other 2. As we assumed prime numbers other than 2, the assumed numbers are odd primes. We know that the sum of two odd numbers is an even number. So, the sum of primes other than 2 is an even number. So, the obtained even number is not a prime. In this case, we can say that the sum of primes cannot be a prime.
Example:
Let us take two prime numbers 3 and 7.
\[3+7=10\]
Sum of 3 and 7 is equal to 10. We know that 10 is an even number and it is not a prime number. So, we can say that the sum of primes cannot be a prime.
Case-2:
Let us assume two prime numbers. Let the first prime number be 2 and the other prime number be an odd prime. We know that the sum of an even number and odd number is an odd number. We know that all the prime numbers other than 2 are odd primes.
Example (a):
Let us take two prime numbers 2 and 5.
\[2+5=7\]
Sum of 2 and 5 is equal to 7. We know that 7 is an odd number and a prime number.
Example (b):
Let us take two prime numbers 2 and 13.
\[2+13=15\]
Sum of 2 and 13 is equal to 15. We know 15 is an odd number and not a prime number.
From example (a) we are having the sum of two prime numbers is a prime number.
From example (b) we are having the sum of prime numbers is not a prime number.
So, collectively we can say that the sum of two prime numbers may be a prime number or may not be a prime number.
Case-3:
Let us assume two prime numbers. Let us assume both the prime numbers as even primes. The one and only even prime number is 2. We know that the sum of 2 and 2 gives 4. We know that 4 is not a prime number.

From the above cases it is clear that the sum of two prime numbers need not be a prime number always.
In the question, we were given that the sum of two prime numbers cannot be a prime number. This statement is incorrect because the sum of two prime numbers may be a prime number which is shown in case 2.
So, option (B) is correct.

Note: We should remember that 1 is neither a prime nor a composite. So, for solving this question don’t consider that 1 is a prime number. If we consider 1 as a prime number then we may get wrong results. We know that every prime number has only 2 factors and every composite number has 3 or more factors. But 1 has only one factor. So, we can prove that 1 is neither a prime nor a composite.