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?

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.

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

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

16{,}383 = 2^{14} - 1

Factor as

(2^{7} - 1)(2^{7} + 1)

We have

16{,}383 = 2^{14} - 1

= (2^{7} - 1)(2^{7} + 1)

= 127 \cdot 129.

Then

129 = 3 \cdot 43,

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.