GATE CS 2025 Set 2 Q5 — General Aptitude

To find the statement that is correct for all values of prime numbers $p_1$ and $p_2$, let us analyze each option:

• Option A: $p_1 + p_2$ is not a prime number
  This is false. If we choose $p_1 = 2$ and $p_2 = 3$, then $p_1 + p_2 = 5$, which is a prime number.
• Option B: $p_1 p_2$ is not a prime number
  This is always true. By definition, a prime number has exactly two distinct positive divisors: $1$ and itself. Since $p_1$ and $p_2$ are prime numbers (where $p_1 \ge 2$ and $p_2 \ge 2$), their product $p_1 p_2$ will have at least the divisors $1, p_1, p_2,$ and $p_1 p_2$. Since $p_1 p_2 > p_1 \ge 2$, the product has divisors other than $1$ and itself, making it a composite number.
• Option C: $p_1 + p_2 + 1$ is a prime number
  This is false. If we choose $p_1 = 3$ and $p_2 = 5$, then $p_1 + p_2 + 1 = 9$, which is composite ($3 \times 3$).
• Option D: $p_1 p_2 + 1$ is a prime number
  This is false. If we choose $p_1 = 3$ and $p_2 = 5$, then $p_1 p_2 + 1 = 16$, which is composite ($2^4$).

Thus, only the statement in Option B is correct for all prime numbers $p_1$ and $p_2$.