Difference between revisions of "2006 AIME I Problems/Problem 9"
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== Problem == | == Problem == | ||
− | The sequence <math> a_1, a_2, \ldots </math> is geometric with <math> a_1=a </math> and common ratio <math> r, </math> where <math> a </math> and <math> r </math> are positive integers. Given that <math> \log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006, </math> find the number of possible ordered pairs <math> (a,r). </math> | + | The [[sequence]] <math> a_1, a_2, \ldots </math> is [[geometric sequence|geometric]] with <math> a_1=a </math> and common [[ratio]] <math> r, </math> where <math> a </math> and <math> r </math> are positive integers. Given that <math> \log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006, </math> find the number of possible ordered pairs <math> (a,r). </math> |
+ | == Solution 1 == | ||
+ | <cmath>\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}) </cmath> | ||
+ | So our question is equivalent to solving <math>\log_8 (a^{12}r^{66})=2006</math> for <math>a, r</math> [[positive integer]]s. <math>a^{12}r^{66}=8^{2006} = (2^3)^{2006} = (2^6)^{1003}</math> so <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 <math>a</math> and <math>r</math> are themselves powers of 2. Now, let <math>a=2^x</math> and <math>r=2^y</math>: | ||
+ | <cmath>\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*}</cmath> | ||
− | == | + | 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>. |
− | <math> | + | |
− | + | ||
− | + | 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> = <math>\boxed{046}</math> to find only the odd solutions. <math>-XxHalo711</math> | |
− | <math>\ | + | |
− | + | -------------- | |
+ | Another way is to write | ||
+ | |||
+ | <math>x = \frac{1003-11y}2</math> | ||
+ | |||
+ | Since <math>1003/11 = 91 + 2/11</math>, the answer is just the number of odd integers in <math>[1,91]</math>, which is, again, <math>\boxed{046}</math>. | ||
+ | ---------------- | ||
+ | |||
+ | == Solution 2 == | ||
+ | Using the above method, we can derive that <math>a^{2}r^{11} = 2^{1003}</math>. | ||
+ | Now, think about what happens when r is an even power of 2. Then <math>a^{2}</math> 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 <math>2^{1}</math>, <math>2^{3}</math>, <math>2^{5}</math> .... all work for r, until r hits <math>2^{93}</math>, when it gets greater than <math>2^{1003}</math>, so the greatest value for r is <math>2^{91}</math>. All that's left is to count the number of odd integers between 1 and 91 (inclusive), which yields <math>\boxed{046}</math>. | ||
+ | |||
+ | == Solution 3 == | ||
+ | Using the method from Solution 1, we get <math>\log_8a^{12}r^{66}=2006 \implies a^{12}r^{66}=8^{2006}=2^{6018}</math>. | ||
+ | |||
+ | Since <math>a</math> and <math>r</math> both have to be powers of <math>2</math>, we can rewrite this as <math>12x+66y=6018</math>. | ||
+ | |||
+ | <math>6018 \equiv 66 \equiv 6\pmod{12}</math>. So, when we subtract <math>12</math> from <math>6018</math>, the result is divisible by <math>66</math>. Evaluating that, we get <math>(1,91)</math> as a valid solution. Since <math>66 \cdot 2 = 12 \cdot 11</math>, when we add <math>11</math> to the value of <math>a</math>, we can subtract <math>2</math> from the value of <math>r</math> to keep the equation valid. Using this, we get <math>(1,91),(12,89),(23,87), \cdots (541,1)</math>. In order to count the number of ordered pairs, we can simply count the number of <math>y</math> values. Every odd number from <math>1</math> to <math>91</math> is included, so we have <math>\boxed{046}</math> solutions. | ||
+ | |||
+ | -Phunsukh Wangdu | ||
== See also == | == See also == | ||
− | + | {{AIME box|year=2006|n=I|num-b=8|num-a=10}} | |
− | [[Category:Intermediate | + | [[Category:Intermediate Algebra Problems]] |
+ | [[Category:Intermediate Number Theory Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 18:27, 22 May 2021
Problem
The sequence is geometric with
and common ratio
where
and
are positive integers. Given that
find the number of possible ordered pairs
Solution 1
So our question is equivalent to solving for
positive integers.
so
.
The product of and
is a power of 2. Since both numbers have to be integers, this means that
and
are themselves powers of 2. Now, let
and
:
For to be an integer, the numerator must be divisible by
. This occurs when
because
. Because only even integers are being subtracted from
, the numerator never equals an even multiple of
. Therefore, the numerator takes on the value of every odd multiple of
from
to
. Since the odd multiples are separated by a distance of
, the number of ordered pairs that work is
. (We must add 1 because both endpoints are being included.) So the answer is
.
For the step above, you may also simply do to find how many multiples of
there are in between
and
. Then, divide
=
to find only the odd solutions.
Another way is to write
Since , the answer is just the number of odd integers in
, which is, again,
.
Solution 2
Using the above method, we can derive that .
Now, think about what happens when r is an even power of 2. Then
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
,
,
.... all work for r, until r hits
, when it gets greater than
, so the greatest value for r is
. All that's left is to count the number of odd integers between 1 and 91 (inclusive), which yields
.
Solution 3
Using the method from Solution 1, we get .
Since and
both have to be powers of
, we can rewrite this as
.
. So, when we subtract
from
, the result is divisible by
. Evaluating that, we get
as a valid solution. Since
, when we add
to the value of
, we can subtract
from the value of
to keep the equation valid. Using this, we get
. In order to count the number of ordered pairs, we can simply count the number of
values. Every odd number from
to
is included, so we have
solutions.
-Phunsukh Wangdu
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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