PK 2015 Q16
16. Which of the following values of \(n\) is a counterexample to the statement, ‘If \(n\) is a prime number, then exactly one of \(n - 2\) and \(n + 2\) is prime’ ?
- A 11
- B 19
- C 21
- D 29
- E 37
Pink Kangaroo 2015, Question 16. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 37
Since 21 is not prime, it cannot give a contradiction to the statement because \(n\) must be prime.
The primes 11, 19, 29 don’t give a contradiction because exactly one of \(n - 2\), \(n + 2\) is a prime for each of them: 11 (9 is not prime, 13 is prime), 19 (17 is prime, 21 is not prime), 29 (27 is not prime, 31 is prime).
However for 37, which is prime, this does give a contradiction.
Solution © UK Mathematics Trust, from the PK 2015 solutions.