Difference between revisions of "1987 AIME Problems/Problem 11"
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== Problem == | == Problem == | ||
| − | Find the largest possible value of <math>\displaystyle k</math> for which <math>\displaystyle 3^{11}</math> is expressible as the sum of <math>\displaystyle k</math> consecutive positive | + | Find the largest possible value of <math>\displaystyle k</math> for which <math>\displaystyle 3^{11}</math> is expressible as the sum of <math>\displaystyle k</math> consecutive [[positive integer]]s. |
== Solution == | == Solution == | ||
| − | {{ | + | Let us write down one such sum, with <math>m</math> terms and first term <math>n + 1</math>: |
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| + | <math>3^{11} = (n + 1) + (n + 2) + \ldots + (n + m) = \frac{1}{2} m(2n + m + 1)</math>. | ||
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| + | Thus <math>m(2n + m + 1) = 2 \cdot 3^{11}</math> so <math>m</math> is a [[divisor]] of <math>2\cdot 3^{11}</math>. However, because <math>n \geq 0</math> we have <math>m^2 < m(m + 1) \leq 2\cdot 3^{11}</math> so <math>m < \sqrt{2\cdot 3^{11}} < 3^6</math>. Thus, we are looking for large factors of <math>2\cdot 3^{11}</math> which are less than <math>3^6</math>. The largest such factor is clearly <math>2\cdot 3^5 = 486</math>; for this value of <math>m</math> we do indeed have the valid [[expression]] <math>3^{11} = 122 + 123 + \ldots + 607</math>. | ||
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== See also == | == See also == | ||
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{{AIME box|year=1987|num-b=10|num-a=12}} | {{AIME box|year=1987|num-b=10|num-a=12}} | ||
| + | [[Category:Intermediate Number Theory Problems]] | ||
Revision as of 19:12, 15 February 2007
Problem
Find the largest possible value of 
for which 
is expressible as the sum of consecutive positive integers.
Solution
Let us write down one such sum, with 
terms and first term 
:
.
Thus 
so is a divisor of 
. However, because 
we have 
so 
. Thus, we are looking for large factors of which are less than 
. The largest such factor is clearly 
; for this value of we do indeed have the valid expression 
.
See also
| 1987 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
