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
GATE 2025 · General Aptitude · Prime Numbers · medium
Answer: B is always correct: p_1 * p_2 is not a prime number for any primes p_1 and p_2.
Eliminate option A with a counterexample: Choose p_1=2 and p_2=3. Then p_1+p_2 = 5, which IS a prime number. So option A ('p_1+p_2 is not a prime') is false for this choice. Option A is eliminated.
Eliminate options C and D with counterexamples: Option C (p_1+p_2+1 always prime): p_1=2, p_2=3 gives 6, which is not prime. Eliminated. Option D (p_1*p_2+1 always prime): p_1=3, p_2=5 gives 16, which is not prime. Eliminated.
Verify option B: p_1 * p_2 is never prime: Since p_1 >= 2 and p_2 >= 2, we have p_1*p_2 >= 4. The number p_1*p_2 has at least three distinct divisors: 1, p_1 (or p_2 if p_1=p_2), and p_1*p_2 itself. A prime has exactly 2 divisors. So p_1*p_2 is always composite (not prime). Option B is always TRUE.