Let both p and k be prime numbers such that (p² + k) is also a prime number less than 30. What is the number of possible values o…

Question

Let both p and k be prime numbers such that (p² + k) is also a prime number less than 30. What is the number of possible values of k?

1. A. 4
2. B. 5 (Correct answer)
3. C. 6
4. D. 7

Correct Answer

Option B — 5

Detailed Solution & Explanation

The correct answer is 5.

Key Points

• p² + k must be prime and < 30. Try p=2: 4+k < 30 → k < 26. k prime: 2→6(not prime), 3→7(prime✓), 5→9(not prime), 7→11(prime✓), 11→15(not), 13→17(prime✓), 17→21(not), 19→23(prime✓).
• Try p=3: 9+k<30→k<21. k=2→11(prime✓). k=3→12(no), k=5→14(no), k=7→16(no), k=11→20(no), k=13→22(no), k=17→26(no), k=19→28(no).
• Try p=5: 25+k<30→k=2→27(no), k=3→28(no).
• So k values: 3,7,13,19,2 = 5 values. Answer: 5.