GK 2019 Q16
16. Chloe chose a three-digit integer with all its digits different and wrote it on lots of pieces of paper. Peter picked some of the pieces of paper and added the three-digit integers on them. His answer was 2331. How many pieces of paper did Peter pick?
- A 2331
- B 21
- C 9
- D 7
- E 3
Grey Kangaroo 2019, Question 16. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 9
First note that the factorisation of 2331 into prime numbers is 2331 = 3 × 3 × 7 × 37. Note also that 3 × 37 = 111. Since Claire’s three-digit integer has all its digits different, and both 3 × 111 = 333 and 7 × 111 = 777 have repeated digits, the factors of Claire’s integer do not include both 3 and 37. However, since 3 × 3 × 7 = 63, which is not a three-digit integer, the factors of Claire’s integer must include 37. Therefore Claire’s integer is 37 × 7 = 259 and hence the number of pieces of paper Peter picked is 2331 ÷ 259 = 9.
Solution © UK Mathematics Trust, from the GK 2019 solutions.