GATE IN 2025 | GA | Question: 5

Question

Let $p_{1}$ and $p_{2}$ denote two arbitrary prime numbers. Which one of the following statements is correct for all values of $p_{1}$ and $p_{2}$ ?

• A. $p_{1}+p_{2}$ is not a prime number.
• B. $p_{1} p_{2}$ is not a prime number.
• C. $p_{1}+p_{2}+1$ is a prime number.
• D. $p_{1} p_{2}+1$ is a prime number.

Answer 1

 

We need to check each option:

1. $p_1+p_2 :$
   $Example:$  $2+3=5$ ,which is prime.
   So, the statement $p_1+p_2$  is not a prime is false.

2. $p_1p_2$:
   Since $p_1, p_2 \geq 2$, their product always has non-trivial factors.( If you multiply two prime numbers, the result will always be bigger and have both of them as factors, so it can’t be prime anymore.)
   Hence,  $p_1p_2 $  is never prime. (Correct Option)

3. $p_1 + p_2 + 1:$
   $Example:$  $3+5+1=9$, which is not prime.
   So, the statement is false.

4. $p_1p_2+1$:
   $Example:$ $ 3\times5+1=16$, which is not prime.
   So, the statement is false.

 

Answer 2

option B)

Answer 3

Option B