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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 $16,383$ ? 
Overseas Competition->Knowledge Point->Number Theory Modules->Prime Factorization
We have $$16383=2^{14}-1=\left(2^{7}+1\right)\left(2^{7}-1\right) =129 \cdot 127 $$. Since $129$ is composite, $127$ is the largest prime divisible by $16383$. The sum of $127$\textquotesingle s digits is $$ 1+2+7=\text { (C) } 10 $$. 