Problem 26 · 2016 Math Kangaroo
Stretch
Number Theory
factor-pairsfactorization
Natural numbers, no two the same, are written on a board. The product of the two smallest is 16, and the product of the two largest is 225. What is the sum of all the numbers on the board?
Show answer
Answer: C — 44
Show hints
Hint 1 of 3
The two smallest numbers multiply to 16; the two largest to 225. Factor each.
Still stuck? Show hint 2 →
Hint 2 of 3
Pick distinct factor pairs that can be the smallest two and largest two, with no room for numbers in between.
Still stuck? Show hint 3 →
Hint 3 of 3
Then add all the numbers.
Show solution
Approach: identify the extreme pairs by factoring
- The two smallest distinct naturals with product 16 are 2 and 8; the two largest distinct with product 225 are 9 and 25.
- Since 8 and 9 are consecutive, no other number can fit between the small and large pairs, so the board holds exactly 2, 8, 9, 25.
- Their sum is 2 + 8 + 9 + 25 = 44.
Mark:
· log in to save