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Difference between revisions of "2006 AIME I Problems/Problem 9"

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For <math>y</math> to be an integer, the [[numerator]] must be [[divisible]] by <math>11</math>.  This occurs when <math>x=1</math> because <math>1001=91*11</math>. Because only [[even integer]]s are being subtracted from <math>1003</math>, the numerator never equals an even [[multiple]] of <math>11</math>. Therefore, the numerator takes on the value of every [[odd integer | odd]] multiple of <math>11</math> from <math>11</math> to <math>1001</math>.  Since the odd multiples are separated by a distance of <math>22</math>, the number of ordered pairs that work is <math>1 + \frac{1001-11}{22}=1 + \frac{990}{22}=46</math>.  (We must add 1 because both endpoints are being included.) So the answer is <math>\boxed{046}</math>.
 
For <math>y</math> to be an integer, the [[numerator]] must be [[divisible]] by <math>11</math>.  This occurs when <math>x=1</math> because <math>1001=91*11</math>. Because only [[even integer]]s are being subtracted from <math>1003</math>, the numerator never equals an even [[multiple]] of <math>11</math>. Therefore, the numerator takes on the value of every [[odd integer | odd]] multiple of <math>11</math> from <math>11</math> to <math>1001</math>.  Since the odd multiples are separated by a distance of <math>22</math>, the number of ordered pairs that work is <math>1 + \frac{1001-11}{22}=1 + \frac{990}{22}=46</math>.  (We must add 1 because both endpoints are being included.) So the answer is <math>\boxed{046}</math>.
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For the step above, you may also simply do <math>1001/11 + 1 = 91 + 1 = 92</math> to find how many multiples of <math>11</math> there are in between <math>11</math> and <math>1001</math>. Then, divide <math>92/2</math> to find only the odd solutions. <math>-XxHalo711</math>
  
 
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Revision as of 23:33, 15 June 2019

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 1

\[\log_8 a_1+\log_8 a_2+\ldots+\log_8 a_{12}= \log_8 a+\log_8 (ar)+\ldots+\log_8 (ar^{11}) \\ = \log_8(a\cdot ar\cdot ar^2\cdot \cdots \cdot ar^{11}) = \log_8 (a^{12}r^{66})\]

So our question is equivalent to solving $\log_8 (a^{12}r^{66})=2006$ for $a, r$ positive integers. $a^{12}r^{66}=8^{2006} = (2^3)^{2006} = (2^6)^{1003}$ so $a^{2}r^{11}=2^{1003}$.

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 themselves powers of 2. Now, let $a=2^x$ and $r=2^y$:

\begin{eqnarray*}(2^x)^2\cdot(2^y)^{11}&=&2^{1003}\\ 2^{2x}\cdot 2^{11y}&=&2^{1003}\\ 2x+11y&=&1003\\ y&=&\frac{1003-2x}{11} \end{eqnarray*}

For $y$ to be an integer, the numerator must be divisible by $11$. This occurs when $x=1$ because $1001=91*11$. Because only even integers are being subtracted from $1003$, the numerator never equals an even multiple of $11$. Therefore, the numerator takes on the value of every odd multiple of $11$ from $11$ to $1001$. Since the odd multiples are separated by a distance of $22$, the number of ordered pairs that work is $1 + \frac{1001-11}{22}=1 + \frac{990}{22}=46$. (We must add 1 because both endpoints are being included.) So the answer is $\boxed{046}$.


For the step above, you may also simply do $1001/11 + 1 = 91 + 1 = 92$ to find how many multiples of $11$ there are in between $11$ and $1001$. Then, divide $92/2$ to find only the odd solutions. $-XxHalo711$

Another way is to write

$x = \frac{1003-11y}2$

Since $1003/11 = 91 + 2/11$, the answer is just the number of odd integers in $[1,91]$, which is, again, $\boxed{046}$.

Solution 2

Using the above method, we can derive that $a^{2}r^{11} = 2^{1003}$. Now, think about what happens when r is an even power of 2. Then $a^{2}$ must be an odd power of 2 in order to satisfy the equation which is clearly not possible. Thus the only restriction r has is that it must be an odd power of 2, so $2^{1}$, $2^{3}$, $2^{5}$ .... all work for r, until r hits $2^{93}$, when it gets greater than $2^{1003}$, so the greatest value for r is $2^{91}$. All that's left is to count the number of odd integers between 1 and 91 (inclusive), which yields $\boxed{046}$.

See also

2006 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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