2021 Fall AMC 10B Problems/Problem 8

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 $16383$ ?

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

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

2021 Fall AMC 10B (Problems • Answer Key • Resources)
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