GK 2020 Q18
18. How many four-digit numbers \(N\) are there, such that half of the number \(N\) is divisible by 2, a third of \(N\) is divisible by 3 and a fifth of \(N\) is divisible by 5?
- A 1
- B 7
- C 9
- D 10
- E 11
Grey Kangaroo 2020, Question 18. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 10
The information in the question tells us that \(N\) is divisible by 2 × 2 = 4, 3 × 3 = 9 and 5 × 5 = 25. Since these three values have no factors in common, \(N\) is divisible by 4 × 9 × 25 = 900. Therefore \(N\) is a four-digit multiple of 900. The smallest such multiple is 2 × 900 = 1800 and the largest is 11 × 900 = 9900. Therefore there are 10 such four-digit numbers.
Solution © UK Mathematics Trust, from the GK 2020 solutions.