Problem:
How many sets of two or more consecutive positive integers have a sum of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
First note that, in general, the sum of consecutive integers is times their median. If the sum is , we have the following cases:
if , then the median is and the two integers are and ;
if , then the median is and the three integers are , and ;
if , then the median is and the five integers are , and .
Because the sum of four consecutive integers is even, cannot be written in such a manner. Also, the sum of more than five consecutive integers must be more than . Hence there are sets satisfying the condition.
Note: It can be shown that the number of sets of two or more consecutive positive integers having a sum of is equal to the number of odd positive divisors of , excluding .
The problems on this page are the property of the MAA's American Mathematics Competitions