PK 2017 Q9
9. How many positive integers \(N\) possess the property that exactly one of the numbers \(N\) and \((N + 20)\) is a 4-digit number?
- A 19
- B 20
- C 38
- D 39
- E 40
Pink Kangaroo 2017, Question 9. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 40
If exactly one of \(N\) and \(N + 20\) has four digits, then the other has either 3 or 5 digits. If \(N\) has 3 digits and \(N + 20\) has 4 digits, then \(980 \leqslant N \leqslant 999\), giving 20 possibilities. If \(N\) has 4 digits and \(N + 20\) has 5 digits, then \(9980 \leqslant N \leqslant 9999\), giving 20 possibilities. Overall there are 40 possibilities for \(N\).
Solution © UK Mathematics Trust, from the PK 2017 solutions.