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2021 Fall AMC 10B Problems/Problem 8

Revision as of 22:17, 22 November 2021 by Kingofpineapplz (talk | contribs) (→‎Solution 1)

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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