Difference between revisions of "2021 Fall AMC 12B Problems/Problem 6"
(Created page with "== 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...") |
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+ | {{duplicate|[[2021 Fall AMC 10B Problems#Problem 8|2021 Fall AMC 10B #8]] and [[2021 Fall AMC 10B Problems#Problem 6|2021 Fall AMC 12B #6]]}} | ||
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== 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 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>? | ||
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<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 1== |
+ | We have | ||
+ | |||
+ | <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>\boxed{\textbf{(C) }10}</math>. | ||
+ | |||
+ | ~kingofpineapplz | ||
+ | |||
+ | ==Solution 2== | ||
+ | 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>. | ||
+ | |||
+ | ~Arcticturn | ||
+ | |||
+ | == Solution 3== | ||
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> | 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> | ||
~NH14 | ~NH14 |
Revision as of 00:14, 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.
Contents
[hide]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 have
Since is prime, our answer is .
~kingofpineapplz
Solution 2
Since is , we can consider it as . is less than , so it can be considered as less than a square. Therefore, it can be expressed as . Since is is . is , and since is larger, our answer is .
~Arcticturn
Solution 3
We want to find the largest prime factor of Thus, the largest prime factor is which has the sum of the digits as Thus the answer is
~NH14