2021 AMC 12B Fall Problem 6

Attempt Problem 6 of the 2021 AMC 12B 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 12B Fall solutions, or check the answer key.

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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:difference of squares prime factorization

Difficulty rating: 1280

Solution:

Since $16{,}383 = 2^{14} - 1$ $= (2^{7} - 1)(2^{7} + 1)$ $= 127 \cdot 129.$ Then $129 = 3 \cdot 43,$ so $16{,}383 = 3 \cdot 43 \cdot 127.$

The greatest prime factor is $127,$ whose digit sum is $1 + 2 + 7 = 10.$

Thus, the correct answer is C.

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