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2006 AIME I Problems/Problem 9: Difference between revisions

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Line 9: Line 9:
<math>\log_8 a_1+\log_8 a_2+\ldots+\log_8 a_{12}=
<math>\log_8 a_1+\log_8 a_2+\ldots+\log_8 a_{12}=
     \log_8 a+log_8 (ar)+\ldots+\log_8 (ar^{11})</math>
     \log_8 a+log_8 (ar)+\ldots+\log_8 (ar^{11})</math>
We must now use the rules of logarithms:


<math>\log_8 a_1+\log_8 a_2+\ldots+\log_8 a_{12}=
<math>\log_8 a_1+\log_8 a_2+\ldots+\log_8 a_{12}=
Line 23: Line 21:
<math>a^{12}r^{66}=(2^3)^{2006}</math>
<math>a^{12}r^{66}=(2^3)^{2006}</math>


<math>a^{12}r^{66}=2^{6018}</math>
<math>a^{2}r^{11}=2^{1003}</math>


<math>a^{2}r^{11}=2^{1003}</math>
The product of <math>a^2</math> and <math>r^{11}</math> is a power of 2.  Since both numbers have to be integers, this means that a and r are also powers of 2.  Now, let <math>a=2^x</math> and <math>r=2^y</math>:


== See also ==
== See also ==

Revision as of 13:57, 3 August 2006

Problem

The sequence $a_1, a_2, \ldots$ is geometric with $a_1=a$ and common ratio $r,$ where $a$ and $r$ are positive integers. Given that $\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006,$ find the number of possible ordered pairs $(a,r).$



Solution

$\log_8 a_1+\log_8 a_2+\ldots+\log_8 a_{12}= \log_8 a+log_8 (ar)+\ldots+\log_8 (ar^{11})$

$\log_8 a_1+\log_8 a_2+\ldots+\log_8 a_{12}= \log_8 (a*ar*ar^2*\cdots*ar^{11})$

$\log_8 a_1+\log_8 a_2+\ldots+\log_8 a_{12}= \log_8 (a^{12}r^{66})$

$\log_8 (a^{12}r^{66})=2006$

$a^{12}r^{66}=8^{2006}$

$a^{12}r^{66}=(2^3)^{2006}$

$a^{2}r^{11}=2^{1003}$

The product of $a^2$ and $r^{11}$ is a power of 2. Since both numbers have to be integers, this means that a and r are also powers of 2. Now, let $a=2^x$ and $r=2^y$:

See also

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