2021 AMC 10B Fall Problem 8

Attempt Problem 8 of the 2021 AMC 10B Fall below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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 $16,383?$

$3$

$7$

$10$

$16$

$22$

Answer: C

Concepts:prime factorization difference of squares

Difficulty rating: 1030

Solution:

We know $\begin{aligned}16,383 &= 16384-1 \\&= 2^{14}-1 \\&= (2^7-1)(2^7+1) \\&= 127\cdot 129.\end{aligned}$ 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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