PK 2019 Q16
16. Sixty apples and sixty pears are to be packed into boxes so that each box contains the same number of apples, and no two boxes contain the same number of pears. What is the largest possible number of boxes that can be packed in this way?
- A 20
- B 15
- C 12
- D 10
- E 6
Pink Kangaroo 2019, Question 16. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 10
Since each box has a different number of pears, and we want as many boxes as possible, we could start by putting no pears in box 1, 1 pear in box 2, 2 pears in box 3, and so on. By the 11th box we have 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55 pears, and there are not enough pears for another box. However, we can’t use 11 boxes because we need to share the 60 apples evenly and 60 is not a multiple of 11. So, the maximum possible is 10 boxes with 6 apples in each box, and the pears distributed as stated but with 24 in the 10th box.
Solution © UK Mathematics Trust, from the PK 2019 solutions.