2021 Fall AMC 12B Problems/Problem 6: Difference between revisions
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== Problem == | == Problem == | ||
The | 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 == | == Solution== | ||
We have | We have | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
| Line 12: | Line 12: | ||
& = \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 \\ | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
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>. | |||
~Steven Chen (www.professorchenedu.com) ~NH14 ~kingofpineapplz ~Arcticturn | ~Steven Chen (www.professorchenedu.com) ~NH14 ~kingofpineapplz ~Arcticturn ~MrThinker ~abed_nadir(youtube.com/@indianmathguy) | ||
~PaperMath | |||
===Video Solution 1== | === 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== | |||
https://youtu.be/p9_RH4s-kBA?t=1121 | |||
==Video Solution (Just 1 min!)== | |||
https://youtu.be/NB6CamKgDaw | https://youtu.be/NB6CamKgDaw | ||
~Education, the Study of Everything | ~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 
is 
because 
. What is the sum of the digits of the greatest prime number that is a divisor of 
?
Solution
We have
Since 
is composite, 
is the largest prime which can divide 
. The sum of 's digits is 
.
~Steven Chen (www.professorchenedu.com) ~NH14 ~kingofpineapplz ~Arcticturn ~MrThinker ~abed_nadir(youtube.com/@indianmathguy) ~PaperMath
Note
Note that you can quickly tell that 
is prime because it is a Mersenne prime.
Video Solution by Interstigation
https://youtu.be/p9_RH4s-kBA?t=1121
Video Solution (Just 1 min!)
~Education, the Study of Everything
Video Solution by WhyMath
~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 | |
These problems are copyrighted © by the Mathematical Association of America. 
