GATE 2025 CS (CS2) – Question 5

General Aptitude · Quantitative Aptitude: Arithmetic and Number Computation · 1 mark · Multiple choice

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.
$p_1p_2$ is not a prime number.
$p_1 + p_2 + 1$ is a prime number.
$p_1p_2 + 1$ is a prime number.

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Correct answer: (B) $p_1p_2$ is not a prime number.

Explanation

A product of two primes has at least the divisors 1, $p_1$, $p_2$ and itself, so it is composite. Counterexamples to the others: 2+3=5, 2+2+1=5 is prime but 3+3+1=7 is prime while 2+7+1=10 is not, and 3·5+1=16 is not prime.