Problem:
Two distinct numbers are selected from the set so that the sum of the remaining numbers is the product of these two numbers. What is the difference of these two numbers?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Answer (E): The sum of the first 37 positive integers is . Let and be the selected numbers. Then . Adding to both sides of the equation and factoring gives . Because , but no other product of positive integers less than 38 equals 704 , it follows that the two numbers are 31 and 21. The requested difference is 10.
The problems on this page are the property of the MAA's American Mathematics Competitions