UPSC Prelims 2025 CSAT, question 27

Topic: Number system

Q27

Let both $p$ and $k$ be prime numbers such that $(p^2+k)$ is also a prime number less than 30. What is the number of possible values of $k$ ?

Solution.

Answer: (b) 5

Since $p$ is prime and $p^2 + k < 30$ with $k \ge 2$ , $p$ can only be $2, 3,$ or $5$ ( $p^2 \in \{4, 9, 25\}$ ).
For $p = 2$ , $4 + k$ is a prime $< 30$ when $k \in \{3, 7, 13, 19\}$ (giving primes $7, 11, 17, 23$ ; note $k = 25$ is not prime).
For $p = 3$ , $9 + k$ is prime only when $k = 2$ (giving $11$ , since for odd $k$ , $9 + k$ is even); for $p = 5$ , $25 + k < 30$ allows $k \in \{2, 3\}$ , giving $27$ and $28$ (neither prime). Thus $k \in \{2, 3, 7, 13, 19\}$ , giving $5$ possible values.

Trap: Do not include $k = 25$ for $p = 2$ ( $4 + 25 = 29$ is prime, but $k = 25$ is composite).