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Difference between revisions of "2021 Fall AMC 12B Problems/Problem 6"

(→‎Solution 3)
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==Solution 1==
 
==Solution 1==
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We have
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We want to find the largest prime factor of
  
 
<cmath>16383=16384-1=2^{14}-1=(2^7+1)(2^7-1)=129\cdot127=3\cdot43\cdot127.</cmath>
 
<cmath>16383=16384-1=2^{14}-1=(2^7+1)(2^7-1)=129\cdot127=3\cdot43\cdot127.</cmath>
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Since <math>127</math> is prime, our answer is <math>\boxed{\textbf{(C) }10}</math>.
 
  
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~kingofpineapplz
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Since <math>127</math> is prime, our answer is <math>1+2+7=\boxed{\textbf{(C) }10}</math>.
  
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==Solution 2==
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~NH14, kingofpineapplz, and Arcticturn
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Since <math>16384</math> is <math>2^14</math>, we can consider it as <math>(2^7)^2</math>. <math>16383</math> is <math>1</math> less than <math>16384</math>, so it can be considered as <math>1</math> less than a square. Therefore, it can be expressed as <math>(x-1)(x+1)</math>. Since <math>2^7</math> is <math>128, 16383</math> is <math>127 \cdot 129</math>. <math>129</math> is <math>3 \cdot 43</math>, and since <math>127</math> is larger, our answer is <math>\boxed {(C) 10}</math>.
 
  
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~Arcticturn
 
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== Solution 3==
 
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We want to find the largest prime factor of <math>2^{14} -1 = (2^7+1)(2^7-1) = (129)(127) = 3 \cdot 43 \cdot 127.</math> Thus, the largest prime factor is <math>127,</math> which has the sum of the digits as <math>10.</math> Thus the answer is <math>\boxed{\textbf{(D.)} \: 10}.</math>
 
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~NH14
 
 
==Video Solution by Interstigation==
 
==Video Solution by Interstigation==
 
https://youtu.be/p9_RH4s-kBA?t=1121
 
https://youtu.be/p9_RH4s-kBA?t=1121
 +
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2021 Fall|ab=B|num-b=7|num-a=9}}
 
{{AMC10 box|year=2021 Fall|ab=B|num-b=7|num-a=9}}
 
{{AMC12 box|year=2021 Fall|ab=B|num-b=5|num-a=7}}
 
{{AMC12 box|year=2021 Fall|ab=B|num-b=5|num-a=7}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:36, 24 November 2021

The following problem is from both the 2021 Fall AMC 10B #8 and 2021 Fall AMC 12B #6, so both problems redirect to this page.

Problem

The largest prime factor of $16384$ is $2$ because $16384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16383$?

$\textbf{(A)} \: 3\qquad\textbf{(B)} \: 7\qquad\textbf{(C)} \: 10\qquad\textbf{(D)} \: 16\qquad\textbf{(E)} \: 22$

Solution 1

We want to find the largest prime factor of

\[16383=16384-1=2^{14}-1=(2^7+1)(2^7-1)=129\cdot127=3\cdot43\cdot127.\]

Since $127$ is prime, our answer is $1+2+7=\boxed{\textbf{(C) }10}$.

~NH14, kingofpineapplz, and Arcticturn

Video Solution by Interstigation

https://youtu.be/p9_RH4s-kBA?t=1121

See Also

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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
2021 Fall AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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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