IMC 2019 Q22
22. Four of the following six-digit integers are always divisible by 7, regardless of the values of the digits \(P\) and \(Q\).
Which of the following is not necessarily a multiple of 7?
- A ‘PQQ PQQ’
- B ‘PQP QPQ’
- C ‘QPQ PQP’
- D ‘PPP PPP’
- E ‘PPP QQQ’
Intermediate Mathematical Challenge 2019, Question 22. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E ‘PPP QQQ’
Note that 1001 = 7 × 143, so that any six-digit number of the form ‘XYZ XYZ’ = 1001 × ‘XYZ’ is necessarily a multiple of 7. So options A and D are both multiples of 7.
Also, 10 101 = 7 × 1443, so that any six-digit number of the form ‘XYX YXY’ = 10101 × ‘XY’ is necessarily a multiple of 7. So options B and C are both multiples of 7.
However, ‘PPP QQQ’ = \(111 \times (1000P + Q) = 3 \times 37 \times (1000P + Q)\) is not necessarily a multiple of 7. For example, when \(P = 1\) and \(Q = 2\), \(1000P + Q = 1002 = 143 \times 7 + 1\).
Solution © UK Mathematics Trust, from the IMC 2019 solutions. Longer versions with investigations: Solutions and investigations (PDF).