GATE CS 2025 Set 2 — Question 5
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General Aptitude → Analytical Aptitude → Numerical Relations & Reasoning
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Question
Let and denote two arbitrary prime numbers. Which one of the following statements is correct for all values of and ?
Correct answer
(B) p₁ p₂ is not a prime number.
Solution
We analyze each option using properties of prime numbers and counterexamples:
1.Option (A): " is not a prime number."
- Counterexample: Let and . Both are prime. Their sum is , which is a prime number. Thus, the statement is false.
- By definition, a prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Since and are primes, and . The product has divisors and . Since , has at least three divisors (or four if ). Therefore, the product of two primes is always composite (not prime). This statement is always true.
- Counterexample: Let and . Their sum plus one is , which is composite (). Thus, the statement is false.
- Counterexample: Let and . The product plus one is , which is composite (). Thus, the statement is false.
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