IMC 2015 Q16
16. You are asked to choose two positive integers, \(m\) and \(n\) with \(m > n\), so that as many as possible of the expressions \(m + n\), \(m - n\), \(m \times n\) and \(m \div n\) have values that are prime.
When you do this correctly, how many of these four expressions have values that are prime?
- A 0
- B 1
- C 2
- D 3
- E 4
Intermediate Mathematical Challenge 2015, Question 16. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 3
All four values cannot be prime. If this were so, both \(m \times n\) and \(m \div n\) would be prime which can happen only if \(m\) is prime and \(n = 1\). If \(m\) is an odd prime then \(m + 1\) is even and at least 4, hence not prime, while if \(m = 2\) then \(m - 1\) is not prime but \(m + 1 = 3\) is. Thus three prime values are the most we can have.
Solution © UK Mathematics Trust, from the IMC 2015 solutions. Longer versions with investigations: Solutions and investigations (PDF).