GATE CS 2025 Set 2 Q.5: Quantitative Aptitude

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 $p_2$ ?

$p_1 + p_2$ is not a prime number.

$p_1 p_2$ is not a prime number.

$p_1 + p_2 + 1$ is a prime number.

$p_1 p_2 + 1$ is a prime number.

Official answer and solution

Official answer: B
From: Official GATE 2025 answer key

Solution

Test small prime pairs and use the definition of a composite number for the product of two primes.

Since $p_1 \ge 2$ and $p_2 \ge 2$ are prime numbers, their product $p_1 p_2$ has $p_1$ and $p_2$ as proper factors ( $1 < p_1 < p_1 p_2$ ), so $p_1 p_2$ is always composite and never prime. For the other options, counterexamples exist:

(A) fails for $p_1 = 2, p_2 = 3$ ( $2 + 3 = 5$ , which is prime).
(C) fails for $p_1 = 3, p_2 = 5$ ( $3 + 5 + 1 = 9 = 3^2$ , not prime).
(D) fails for $p_1 = 3, p_2 = 5$ ( $3 \times 5 + 1 = 16 = 2^4$ , not prime).

Final answer

Option (B)

Common trap

Assuming all prime numbers are odd and concluding that $p_1 + p_2$ is always even and composite (forgetting $p = 2$ ).

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 2 (CS2)
Question: 5 of 65
Section: General Aptitude

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