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Difference between revisions of "2019 AMC 10A Problems/Problem 18"

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==Solution 1==
 
==Solution 1==
  
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We can expand the fraction <math>0.\overline{23}_k</math> as follows: <math>0.\overline{23}_k = 2\cdot k^{-1} + 3 \cdot k^{-2} + 2 \cdot k^{-3} + 3 \cdot k^{-4} + ...</math>. Notice that this is equivalent to  
+
We can expand the fraction <math>0.\overline{23}_k</math> as follows: <math>0.\overline{23}_k = 2\cdot k^{-1} + 3 \cdot k^{-2} + 2 \cdot k^{-3} + 3 \cdot k^{-4} + ...</math> Notice that this is equivalent to  
 
<cmath>2( k^{-1} + k^{-3} + k^{-5} + ... ) + 3 (k^{-2} + k^{-4} + k^{-6} + ... )</cmath>
 
<cmath>2( k^{-1} + k^{-3} + k^{-5} + ... ) + 3 (k^{-2} + k^{-4} + k^{-6} + ... )</cmath>
  
−
By summing the infinite series and simplifying, we have <math>\frac{2k+3}{k^2-1} = \frac{7}{51}</math>. Solving this quadratic equation or testing the answer choices yields the answer <math>\boxed{k=16}.</math>
+
By summing the geometric series and simplifying, we have <math>\frac{2k+3}{k^2-1} = \frac{7}{51}</math>. Solving this quadratic equation (or simply testing the answer choices) yields the answer <math>k = \boxed{\textbf{(D) }16}</math>.
  
 
==Solution 2==
 
==Solution 2==
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Let <math>a = 0.2323\dots_k</math>. Therefore, <math>k^2a=23.2323\dots_k</math>.
 
Let <math>a = 0.2323\dots_k</math>. Therefore, <math>k^2a=23.2323\dots_k</math>.
  
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From this, we see that <math>k^2a-a=23_k</math>.  
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From this, we see that <math>k^2a-a=23_k</math>, so <math>a = \frac{23_k}{k^2-1} = \frac{2k+3}{k^2-1} = \frac{7}{51}</math>.
  
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Solving for a:
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Now, similar to in Solution 1, we can either test if <math>2k+3</math> is a multiple of 7 with the answer choices, or actually solve the quadratic, so that the answer is <math>\boxed{\textbf{(D) }16}</math>.
  
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<math>a = \frac{23_k}{k^2-1} = \frac{2k+3}{k^2-1} = \frac{7}{51}</math>
+
==Solution 3 (bash)==
 +
We can simply plug in all the answer choices as values of <math>k</math>, and see which one works. After legendary, amazingly, historically great calculations, this eventually gives us <math>\boxed{\textbf{(D) }16}</math> as the answer.
  
−
Similar to Solution 1, testing if <math>2k+3</math> is a multiple of 7 with the answer choices or solving the quadratic yields <math>k=16</math>, so the answer is <math>\boxed{D}</math>.
+
-ellpet
  
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==Solution 3 (extremely rigorous)==
+
==Solution 4==
−
Plug in the values of k and bash. This gives us <math>\boxed{D}</math>.
+
Just as in Solution 1, we arrive at the equation <math>\frac{2k+3}{k^2-1}=\frac{7}{51}</math>.
 +
 
 +
We can now rewrite this as <math>\frac{2k+3}{(k-1)(k+1)}=\frac{7}{51}=\frac{7}{3\cdot 17}</math>. Notice that <math>2k+3=2(k+1)+1=2(k-1)+5</math>. As <math>17</math> is a prime, we therefore must have that one of <math>k-1</math> and <math>k+1</math> is divisible by <math>17</math>. Now, checking each of the answer choices, this gives <math>\boxed{\textbf{(D) }16}</math>.
 +
 
 +
==Solution 5==
 +
Assuming you are familiar with the rules for basic repeating decimals, <math>0.232323... = \frac{23}{99}</math>. Now we want our base, <math>k</math>, to conform to <math>23\equiv7\pmod k</math> and <math>99\equiv51\pmod k</math>, the reason being that we wish to convert the number from base <math>10</math> to base <math>k</math>. Given the first equation, we know that <math>k</math> must equal 9, 16, 23, or generally, <math>7n+2</math>. The only number in this set that is one of the multiple choices is <math>16</math>. When we test this on the second equation, <math>99\equiv51\pmod k</math>, it comes to be true. Therefore, our answer is <math>\boxed{\textbf{(D) }16}</math>.
 +
 
 +
==Solution 6==
  
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==Solution 4==
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Note that the LHS equals <cmath>\bigg(\frac{2}{k} + \frac{2}{k^3} + \cdots \bigg) + \bigg(\frac{3}{k^2} + \frac{3}{k^4} + \cdots \bigg) = \frac{\frac{2}{k}}{1 - \frac{1}{k^2}} + \frac{\frac{3}{k^2}}{1 - \frac{1}{k^2}} = \frac{2k+3}{k^2},</cmath> from which we see our equation becomes <cmath>\frac{2k+3}{k^2-1} = \frac{7}{51}, \ \ \implies \ \ 51(2k+3) = 7(k^2-1).</cmath>  
−
Similar to Solution 1, we arrive at <math>\frac{2k+3}{k^2-1}=\frac{7}{51}</math>. We can rewrite this as <math>\frac{2k+3}{(k-1)(k+1)}=\frac{7}{51}=\frac{7}{3\cdot 17}</math>. Notice that <math>2k+3=2(k+1)+1=2(k-1)+5</math>. As <math>17</math> is a prime, we have that one of <math>k-1</math> and <math>k+1</math> is divisible by <math>17</math>. Looking at the answer choices, this gives <math>\boxed{\textbf{(D) }16}</math>.
+
 
 +
Note that <math>17</math> therefore divides <math>k^2 - 1,</math> but as <math>17</math> is prime this therefore implies <cmath>k \equiv \pm 1 \pmod{17}.</cmath> (Warning: This would not be necessarily true if <math>17</math> were composite.) Note that <math>\boxed{\textbf{(D)} 16 }</math> is the only answer choice congruent satisfying this modular congruence, thus completing the problem. <math>\square</math>
 +
 
 +
~ Professor-Mom
 +
 
 +
==Video Solution==
 +
For those who want a video solution: https://www.youtube.com/watch?v=DFfRJolhwN0
  
 
==Video Solution==
 
==Video Solution==
−
For those who want a video solution : https://www.youtube.com/watch?v=DFfRJolhwN0
+
https://youtu.be/3YhYGSneu70
 +
 
 +
Education, the Study of Everything
 +
 
 +
(Please put video solutions at the end in order of when they were edited in)
 +
==Video Solution by TheBeautyofMath==
 +
https://youtu.be/FFpBMkKnOk8?t=1263
  
 +
~IceMatrix
 
==See Also==
 
==See Also==
  
 
{{AMC10 box|year=2019|ab=A|num-b=17|num-a=19}}
 
{{AMC10 box|year=2019|ab=A|num-b=17|num-a=19}}
 
{{AMC12 box|year=2019|ab=A|num-b=10|num-a=12}}
 
{{AMC12 box|year=2019|ab=A|num-b=10|num-a=12}}
 +
[[Category:Introductory Number Theory Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 07:04, 19 February 2021

The following problem is from both the 2019 AMC 10A #18 and 2019 AMC 12A #11, so both problems redirect to this page.

Problem

For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$?

$\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17$

Solution 1

We can expand the fraction $0.\overline{23}_k$ as follows: $0.\overline{23}_k = 2\cdot k^{-1} + 3 \cdot k^{-2} + 2 \cdot k^{-3} + 3 \cdot k^{-4} + ...$ Notice that this is equivalent to \[2( k^{-1} + k^{-3} + k^{-5} + ... ) + 3 (k^{-2} + k^{-4} + k^{-6} + ... )\]

By summing the geometric series and simplifying, we have $\frac{2k+3}{k^2-1} = \frac{7}{51}$. Solving this quadratic equation (or simply testing the answer choices) yields the answer $k = \boxed{\textbf{(D) }16}$.

Solution 2

Let $a = 0.2323\dots_k$. Therefore, $k^2a=23.2323\dots_k$.

From this, we see that $k^2a-a=23_k$, so $a = \frac{23_k}{k^2-1} = \frac{2k+3}{k^2-1} = \frac{7}{51}$.

Now, similar to in Solution 1, we can either test if $2k+3$ is a multiple of 7 with the answer choices, or actually solve the quadratic, so that the answer is $\boxed{\textbf{(D) }16}$.

Solution 3 (bash)

We can simply plug in all the answer choices as values of $k$, and see which one works. After legendary, amazingly, historically great calculations, this eventually gives us $\boxed{\textbf{(D) }16}$ as the answer.

-ellpet

Solution 4

Just as in Solution 1, we arrive at the equation $\frac{2k+3}{k^2-1}=\frac{7}{51}$.

We can now rewrite this as $\frac{2k+3}{(k-1)(k+1)}=\frac{7}{51}=\frac{7}{3\cdot 17}$. Notice that $2k+3=2(k+1)+1=2(k-1)+5$. As $17$ is a prime, we therefore must have that one of $k-1$ and $k+1$ is divisible by $17$. Now, checking each of the answer choices, this gives $\boxed{\textbf{(D) }16}$.

Solution 5

Assuming you are familiar with the rules for basic repeating decimals, $0.232323... = \frac{23}{99}$. Now we want our base, $k$, to conform to $23\equiv7\pmod k$ and $99\equiv51\pmod k$, the reason being that we wish to convert the number from base $10$ to base $k$. Given the first equation, we know that $k$ must equal 9, 16, 23, or generally, $7n+2$. The only number in this set that is one of the multiple choices is $16$. When we test this on the second equation, $99\equiv51\pmod k$, it comes to be true. Therefore, our answer is $\boxed{\textbf{(D) }16}$.

Solution 6

Note that the LHS equals \[\bigg(\frac{2}{k} + \frac{2}{k^3} + \cdots \bigg) + \bigg(\frac{3}{k^2} + \frac{3}{k^4} + \cdots \bigg) = \frac{\frac{2}{k}}{1 - \frac{1}{k^2}} + \frac{\frac{3}{k^2}}{1 - \frac{1}{k^2}} = \frac{2k+3}{k^2},\] from which we see our equation becomes \[\frac{2k+3}{k^2-1} = \frac{7}{51}, \ \ \implies \ \ 51(2k+3) = 7(k^2-1).\]

Note that $17$ therefore divides $k^2 - 1,$ but as $17$ is prime this therefore implies \[k \equiv \pm 1 \pmod{17}.\] (Warning: This would not be necessarily true if $17$ were composite.) Note that $\boxed{\textbf{(D)} 16 }$ is the only answer choice congruent satisfying this modular congruence, thus completing the problem. $\square$

~ Professor-Mom

Video Solution

For those who want a video solution: https://www.youtube.com/watch?v=DFfRJolhwN0

Video Solution

https://youtu.be/3YhYGSneu70

Education, the Study of Everything

(Please put video solutions at the end in order of when they were edited in)

Video Solution by TheBeautyofMath

https://youtu.be/FFpBMkKnOk8?t=1263

~IceMatrix

See Also

2019 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2019 AMC 12A (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 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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