Difference between revisions of "2021 Fall AMC 12B Problems/Problem 6"
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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 by Interstigation== | + | === 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 | 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.
Contents
[hide]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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.