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2021 Fall AMC 12B Problem 6

Problem 6 of 25EasierAlgebraNumber Theory

The greatest prime number that is a divisor of 16,38416{,}384 is 22 because 16,384=214.16{,}384 = 2^{14}. What is the sum of the digits of the greatest prime number that is a divisor of 16,383?16{,}383?

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Solution

We have 16,383=214−116{,}383 = 2^{14} - 1 =(27−1)(27+1)= (2^{7} - 1)(2^{7} + 1) =127⋅129.= 127 \cdot 129. Then 129=3⋅43,129 = 3 \cdot 43, so 16,383=3⋅43⋅127.16{,}383 = 3 \cdot 43 \cdot 127. The greatest prime factor is 127,127, whose digit sum is 1+2+7=10.1 + 2 + 7 = 10. Thus, the correct answer is C.

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Concepts: difference of squares · prime factorization

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.