2021 Fall AMC 12B Problems/Problem 6

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

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

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2021 Fall AMC 12B Problems/Problem 6

Problem

Solution