2021 AMC 10B Fall Problem 8

Question

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

16383=2^{14}-1

Use

2^{14}-1=(2^7-1)(2^7+1)

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 .

Problem 8 in other years