Problem:
In how many ways can 345 be written as the sum of an increasing sequence of two or more consecutive positive integers?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
A sum of consecutive integers is equal to the number of integers in the sum multiplied by their median. Note that . If there are an odd number of integers in the sum, then the median and the number of integers must be complementary factors of 345 . The only possibilities are 3 integers with median integers with median integers with median 23 , and 23 integers with median . Having more integers in the sum would force some of the integers to be negative. If there are an even number of integers in the sum, say , then the median will be , where and are complementary factors of 345 . The possibilities are 2 integers with median integers with median , and 10 integers with median . Again, having more integers in the sum would force some of the integers to be negative. This gives a total of solutions.
The problems on this page are the property of the MAA's American Mathematics Competitions