Problem:
Two distinct numbers are selected from the set {1,2,3,4,…,36,37} so that the sum of the remaining 35 numbers is the product of these two numbers. What is the difference of these two numbers?
Answer Choices:
A. 5
B. 7
C. 8
D. 9
E. 10
Solution:
The sum of the first n integers is given by 2n(n+1), so 237(37+1)=703. Therefore,
703−x−y=xy
Rearranging, xy+x+y=703. We can factor this equation by SFFT to get
(x+1)(y+1)=704
Looking at the possible divisors of 704=26⋅11,22 and 32 are within the constraints of 0<x≤y≤37 so we try those:
(x+1)(y+1)x+1=22,x=21,=22⋅32 y+1=32 y=31
Therefore, the difference y−x=31−21=(E) 10 .
The problems on this page are the property of the MAA's American Mathematics Competitions