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 $16383$ ?

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

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

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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2021 Fall AMC 10B Problems/Problem 8

Contents

Problem 8

Solution 1

Solution 2

See Also