Difference between revisions of "2021 Fall AMC 12B Problems/Problem 6"
Ehuang0531 (talk | contribs) (all 3 solutions are the same) |
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~NH14, kingofpineapplz, and Arcticturn | ~NH14, kingofpineapplz, and Arcticturn | ||
+ | |||
+ | == Solution 2 == | ||
+ | We have | ||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | 16,383 & = 2^{14} - 1 \ | ||
+ | & = \left( 2^7 + 1 \right) \left( 2^7 - 1 \right) \ | ||
+ | & = 129 \cdot 127 \ | ||
+ | & = 3 \cdot 43 \cdot 127 . | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | |||
+ | Therefore, the greatest prime divisor of 16.383 is 127. | ||
+ | |||
+ | Therefore, the answer is <math>\boxed{\textbf{(C) }10}</math>. | ||
+ | |||
+ | ~Steven Chen (www.professorchenedu.com) | ||
==Video Solution by Interstigation== | ==Video Solution by Interstigation== |
Revision as of 21:22, 25 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 is because . What is the sum of the digits of the greatest prime number that is a divisor of ?
Solution 1
We want to find the largest prime factor of
Since is prime, our answer is .
~NH14, kingofpineapplz, and Arcticturn
Solution 2
We have
Therefore, the greatest prime divisor of 16.383 is 127.
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
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.