Computer Sciences > GATE 2025 SET-2 > Number Theory
Let p1 and p2 denote two arbitrary prime numbers. Which one of the following statements is correct for all values of p1 and p2?
Correct : b
Let p₁ and p₂ be two arbitrary prime numbers.
Option (a) states that p₁ + p₂ is not a prime number — this is false because, for example, 2 + 5 = 7, which is prime.
Option (b) states that p₁ × p₂ is not a prime number — this is always true, because the product of two primes is always composite (it has factors other than 1 and itself).
Option (c) states that p₁ + p₂ + 1 is a prime number — this is not always true as there is no guarantee.
Option (d) states that p₁ × p₂ + 1 is a prime number — also not always true.
✅ Final Answer: Option (b) — p₁ × p₂ is not a prime number.
Reason: The product of two prime numbers is always composite, because it has at least the two prime factors p₁ and p₂.
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