Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2021 Fall AMC 12B Problems/Problem 6"
m (→‎Solution)
 
(19 intermediate revisions by 9 users not shown)
Line 2: Line 2:
  
 
== Problem ==
 
== Problem ==
−
The largest prime factor of <math>16384</math> is <math>2</math> because <math>16384 = 2^{14}</math>. What is the sum of the digits of the greatest prime number that is a divisor of <math>16383</math>?
+
The greatest prime number that is a divisor of <math>16{,}384</math> is <math>2</math> because <math>16{,}384 = 2^{14}</math>. What is the sum of the digits of the greatest prime number that is a divisor of <math>16{,}383</math>?
  
 
<math>\textbf{(A)} \: 3\qquad\textbf{(B)} \: 7\qquad\textbf{(C)} \: 10\qquad\textbf{(D)} \: 16\qquad\textbf{(E)} \: 22</math>
 
<math>\textbf{(A)} \: 3\qquad\textbf{(B)} \: 7\qquad\textbf{(C)} \: 10\qquad\textbf{(D)} \: 16\qquad\textbf{(E)} \: 22</math>
  
−
==Solution 1==
+
== Solution==
−
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>
 
−
 
 
−
Since <math>127</math> is prime, our answer is <math>1+2+7=\boxed{\textbf{(C) }10}</math>.
 
−
 
 
−
~NH14, kingofpineapplz, and Arcticturn
 
−
 
 
−
== Solution 2 ==
 
 
We have
 
We have
−
<cmath>
+
<cmath>\begin{align*}
−
\begin{align*}
+
16383 & = 2^{14} - 1 \\
−
16,383 & = 2^{14} - 1 \\
 
 
& = \left( 2^7 + 1 \right) \left( 2^7 - 1 \right) \\
 
& = \left( 2^7 + 1 \right) \left( 2^7 - 1 \right) \\
 
& = 129 \cdot 127 \\
 
& = 129 \cdot 127 \\
−
& = 3 \cdot 43 \cdot 127 .
+
\end{align*}</cmath>
−
\end{align*}
 
−
</cmath>
 
  
−
Therefore, the greatest prime divisor of 16.383 is 127.
+
Since <math>129</math> is composite, <math>127</math> is the largest prime which can divide <math>16383</math>. The sum of <math>127</math>'s digits is <math>1+2+7=\boxed{\textbf{(C) }10}</math>.
  
−
Therefore, the answer is <math>\boxed{\textbf{(C) }10}</math>.
+
~Steven Chen (www.professorchenedu.com) ~NH14 ~kingofpineapplz ~Arcticturn ~MrThinker ~abed_nadir(youtube.com/@indianmathguy)
 +
~PaperMath
  
−
~Steven Chen (www.professorchenedu.com)
+
=== Note ===
 +
Note that you can quickly tell that <math>2^7 -1</math> is prime because it is a [[Mersenne prime|Mersenne prime]].
 +
 
 +
~[https://artofproblemsolving.com/wiki/index.php/User:South South]
  
 
==Video Solution by Interstigation==
 
==Video Solution by Interstigation==
 
https://youtu.be/p9_RH4s-kBA?t=1121
 
https://youtu.be/p9_RH4s-kBA?t=1121
 +
 +
==Video Solution (Just 1 min!)==
 +
https://youtu.be/NB6CamKgDaw
 +
 +
~Education, the Study of Everything
 +
 +
==Video Solution by WhyMath==
 +
https://youtu.be/JBNbjKsw_tU
 +
 +
~savannahsolver
 +
==Video Solution by TheBeautyofMath==
 +
For AMC 10: https://www.youtube.com/watch?v=RyN-fKNtd3A&t=797
 +
 +
For AMC 12: https://www.youtube.com/watch?v=4qgYrCYG-qw
 +
 +
~IceMatrix
  
 
==See Also==
 
==See Also==

Latest revision as of 00:54, 3 November 2024

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 greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16{,}383$?

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

Solution

We have \begin{align*} 16383 & = 2^{14} - 1 \\ & = \left( 2^7 + 1 \right) \left( 2^7 - 1 \right) \\ & = 129 \cdot 127 \\ \end{align*}

Since $129$ is composite, $127$ is the largest prime which can divide $16383$. The sum of $127$'s digits is $1+2+7=\boxed{\textbf{(C) }10}$.

~Steven Chen (www.professorchenedu.com) ~NH14 ~kingofpineapplz ~Arcticturn ~MrThinker ~abed_nadir(youtube.com/@indianmathguy) ~PaperMath

Note

Note that you can quickly tell that $2^7 -1$ is prime because it is a Mersenne prime.

~South

Video Solution by Interstigation

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

Video Solution (Just 1 min!)

https://youtu.be/NB6CamKgDaw

~Education, the Study of Everything

Video Solution by WhyMath

https://youtu.be/JBNbjKsw_tU

~savannahsolver

Video Solution by TheBeautyofMath

For AMC 10: https://www.youtube.com/watch?v=RyN-fKNtd3A&t=797

For AMC 12: https://www.youtube.com/watch?v=4qgYrCYG-qw

~IceMatrix

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_Fall_AMC_12B_Problems/Problem_6&oldid=231313"