Problem 22 · 2019 Math Kangaroo
Hard
Number Theory
divisibilitysum-constraint
60 apples and 60 pears in total are shared out among several boxes. There should be the same number of apples in each box, but no two boxes should contain the same number of pears. Each box contains both fruits. What is the maximum number of boxes that can be filled in this way?
Show answer
Answer: D — 10
Show hints
Hint 1 of 3
Equal apples per box means the number of boxes divides 60.
Still stuck? Show hint 2 →
Hint 2 of 3
Distinct positive pear counts summing to 60 need at least 1+2+…+k; combine both limits.
Still stuck? Show hint 3 →
Hint 3 of 3
Find the largest k with 60 divisible by k and 1+2+…+k ≤ 60.
Show solution
Approach: combine the apple and pear constraints
- With k boxes, k must divide 60 (equal apples) and 1+2+…+k ≤ 60 (distinct pear counts).
- 1+…+10 = 55 ≤ 60 and 10 divides 60, so k = 10 works.
- k = 12 fails (1+…+12 = 78 > 60), so the maximum is 10 boxes.
Mark:
· log in to save