1964 AHSME Problems/Problem 28
Problem
The sum of terms of an arithmetic progression is , and the common difference is . If the first term is an integer, and , then the number of possible values for is:
Solution
Let the progression start at , have common difference , and end at .
The average term is , or . Since the number of terms is , and the sum of the terms is , we have:
Since is a positive integer, it must be a factor of . This means are the only possibilities. We are given , leaving the other five factors.
We now must check if is an integer. We have . If is a factor of , then will be an integer. Adding wil keep it an integer.
Thus, there are possible values for , which is answer .
See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
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