GK 2016 Q21
21. Several different positive integers are written on a blackboard. The product of the smallest two of them is 16. The product of the largest two of them is 225. What is the sum of all the integers written on the blackboard?
- A 38
- B 42
- C 44
- D 58
- E 243
Grey Kangaroo 2016, Question 21. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 44
The only ways to express 16 as the product of two different positive integers are 1 × 16 and 2 × 8. The only ways to express 225 as the product of two different positive integers are 1 × 225, 3 × 75, 5 × 45 and 9 × 25. Therefore, since both integers in the first pair must be smaller than both integers in the second pair, the only possible combination is for the two smallest integers to be 2 and 8 and for the two largest integers to be 9 and 25. There are no other integers written on the blackboard since they would need to be different from these four, be less than 9 and greater than 8. Hence the sum of the integers written on the blackboard is 2 + 8 + 9 + 25 = 44.
Solution © UK Mathematics Trust, from the GK 2016 solutions.