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Difference between revisions of "2021 Fall AMC 10B Problems/Problem 8"

(Created page with "==Problem 8== 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 largest prime factor...")
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Revision as of 22:16, 22 November 2021

Problem 8

The largest prime factor of $16384$ is $2$, because $16384 = 2^{14}$. What is the sum of the digits of the largest prime factor 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

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