IMC 2014 Q4
4. You are given that \(m\) is an even integer and \(n\) is an odd integer. Which of these is an odd integer ?
- A \(3m + 4n\)
- B \(5mn\)
- C \((m + 3n)^2\)
- D \(m^3n^3\)
- E \(5m + 6n\)
Intermediate Mathematical Challenge 2014, Question 4. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C \((m + 3n)^2\)
Since \(m\) is even, \(m = 2k\) for some integer \(k\). So \(3m + 4n = 2(3k + 2n)\); \(5mn = 2(5kn)\); \(m^3n^3 = 8k^3n^3\) and \(5m + 6n = 2(5k + 3n)\), which are all even. As \(n\) is odd, \(3n\) is also odd. So \(m + 3n\) is an even integer plus an odd integer and is therefore odd. The square of an odd integer is odd so \((m + 3n)^2\) is odd.
Solution © UK Mathematics Trust, from the IMC 2014 solutions. Longer versions with investigations: Solutions and investigations (PDF).