IMC 2017 Q9
9. The number ‘tu’ is the two-digit number with units digit \(u\) and tens digit \(t\). The digits \(a\) and \(b\) are distinct, and non-zero. What is the largest possible value of ‘ab’ − ‘ba’?
- A 81
- B 72
- C 63
- D 54
- E 45
Intermediate Mathematical Challenge 2017, Question 9. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 72
First note that as ‘ab’ and ‘ba’ are two digit numbers, neither \(a\) nor \(b\) is equal to zero. Now ‘ab’ − ‘ba’ = \((10a + b) - (10b + a) = 9a - 9b = 9(a - b)\). So for the difference to be as large as possible, \(a\) must be as large as possible, that is 9, and \(b\) must be as small as possible, that is 1.
So the required difference is 91 − 19 = 72.
Solution © UK Mathematics Trust, from the IMC 2017 solutions. Longer versions with investigations: Solutions and investigations (PDF).