Problem:
Fifteen integers are arranged in order on a number line. The integers are equally spaced and have the property that
What is the sum of the digits of
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let be the common difference. If we let and , we see that . Since all the numbers are integers, must be at least . Also, if and , we get that . Once again, since all the numbers are integers, is at most . This tells us that . Note that .
This means that must be at least for to be within the desired range. If , however, becomes greater than , which is not allowed. Now we know that and . This tells us that .
Therefore, the sum of the digits is .
Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions