2021 AMC 10B Fall Problem 8

Below is the professionally curated solution for Problem 8 of the 2021 AMC 10B Fall, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 10B Fall solutions, or check the answer key.

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Concepts:prime factorization difference of squares

Difficulty rating: 1030

The greatest prime number that is a divisor of $16,384$ is $2$ because $16,384 = 2^{14}.$ What is the sum of the digits of the greatest prime number that is a divisor of $16,383?$

$3$

$7$

$10$

$16$

$22$

Solution:

We know $\begin{align*}16,383 &= 16384-1 \\&= 2^{14}-1 \\&= (2^7-1)(2^7+1) \\&= 127\cdot 129.\end{align*}$ Since $129=3\cdot 43,$ we get $16383 = 3\cdot 43\cdot 127.$ Therefore, $127$ is the largest prime factor, and the sum of its digits is $10.$

Thus, the answer is C .

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