2021 Fall AMC 12B Problem 6

Problem 6 of 25EasierAlgebra Number Theory

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?

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Solution

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.

Tagged: difference of squares · prime factorization