Primes p,k where p-squared+k is prime under 30
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?
Options
4
5
6
7
Explanation
We test prime values for p keeping the constraint p² + k < 30 in mind. Case 1: p = 2, so p² = 4. We need 4 + k to be prime. Testing primes for k: 4+2=6 (no); 4+3=7 (yes); 4+5=9 (no); 4+7=11 (yes); 4+11=15 (no); 4+13=17 (yes); 4+17=21 (no); 4+19=23 (yes); 4+23=27 (no). Valid k: {3, 7, 13, 19}. Case 2: p = 3, so p² = 9. We need 9 + k to be prime. Since 9 is odd, an odd ⟨MATH⟩k⟨/MATH⟩ gives an even number > 2 (composite). Thus, ⟨MATH⟩k⟨/MATH⟩ must be the only even prime, 2. 9+2=11 (yes). Valid k: {2}. Case 3: p = 5, p² = 25. k must be 2. 25+2=27 (composite). No valid k. The distinct valid values for k are {2, 3, 7, 13, 19}. There are exactly 5 values.
Answer: (b).
Question details
Year
2025
Paper
CSAT
Question
Q37
Section
Numerical Ability
Sub-topic
Prime Numbers
Type
Number theory
Difficulty
Hard
Source hint
Number theory
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