Difference between revisions of "1964 AHSME Problems/Problem 28"
Talkinaway (talk | contribs) (Created page with "== Problem== The sum of <math>n</math> terms of an arithmetic progression is <math>153</math>, and the common difference is <math>2</math>. If the first term is an integer,...") |
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<math>n(a+n-1) = 153</math> | <math>n(a+n-1) = 153</math> | ||
− | Since <math>n</math> is a positive integer, it must be a factor of <math>153</math>. This means <math>n = 1, 3, 9, 17, 51, 153</math> are the only possibilities. We are given <math>n>1</math>. | + | Since <math>n</math> is a positive integer, it must be a factor of <math>153</math>. This means <math>n = 1, 3, 9, 17, 51, 153</math> are the only possibilities. We are given <math>n>1</math>, leaving the other five factors. |
We now must check if <math>a</math> is an integer. We have <math>a = \frac{153}{n} + 1 - n</math>. If <math>n</math> is a factor of <math>153</math>, then <math>\frac{153}{n}</math> will be an integer. Adding <math>1-n</math> wil keep it an integer. | We now must check if <math>a</math> is an integer. We have <math>a = \frac{153}{n} + 1 - n</math>. If <math>n</math> is a factor of <math>153</math>, then <math>\frac{153}{n}</math> will be an integer. Adding <math>1-n</math> wil keep it an integer. |
Latest revision as of 22:04, 24 July 2019
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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All AHSME Problems and Solutions |
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