PK 2017 Q19
19. Maria chooses two digits \(a\) and \(b\) and uses them to make a six-digit number \(ababab\). Which of the following is always a factor of numbers formed in this way?
- A 2
- B 5
- C 7
- D 9
- E 11
Pink Kangaroo 2017, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 7
Let ‘\(ab\)’ be the 2-digit number with digits \(a\) and \(b\), then the 6-digit number ‘\(ababab\)’ = ‘\(ab\)’ × 10101 = ‘\(ab\)’ × 3 × 7 × 13 × 37 so is always divisible by 3, 7, 13 and 37. But it is only divisible by 2, 5, 9 or 11 if ‘\(ab\)’ is.
Solution © UK Mathematics Trust, from the PK 2017 solutions.