Difference between revisions of "2021 Fall AMC 12B Problems/Problem 6"

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.

Problem

The largest prime factor of $16384$ is $2$ because $16384 = 2^{14}$ . What is the sum of the digits of the greatest prime number that is a divisor of $16383$ ?

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

Solution 1

We have

$16383=16384-1=2^{14}-1=(2^7+1)(2^7-1)=129\cdot127=3\cdot43\cdot127.$ Since $127$ is prime, our answer is $\boxed{\textbf{(C) }10}$ .

~kingofpineapplz

Solution 2

Since $16384$ is $2^14$ , we can consider it as $(2^7)^2$ . $16383$ is $1$ less than $16384$ , so it can be considered as $1$ less than a square. Therefore, it can be expressed as $(x-1)(x+1)$ . Since $2^7$ is $128, 16383$ is $127 \cdot 129$ . $129$ is $3 \cdot 43$ , and since $127$ is larger, our answer is $\boxed {(C) 10}$ .

~Arcticturn

Solution 3

We want to find the largest prime factor of $2^{14} -1 = (2^7+1)(2^7-1) = (129)(127) = 3 \cdot 43 \cdot 127.$ Thus, the largest prime factor is $127,$ which has the sum of the digits as $10.$ Thus the answer is $\boxed{\textbf{(D.)} \: 10}.$

~NH14

@@ Line 1: / Line 1: @@
+{{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]]}}
 == 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>?
@@ Line 4: / Line 6: @@
 <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>
 ~NH14

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