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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 $$ \begin{aligned} 16383 \& =2^{14}-1 \textbackslash\textbackslash{} \& =\left(2^{7}+1\right)\left(2^{7}-1\right) \textbackslash\textbackslash{} \& =129 \cdot 127 \end{aligned} $$ Since 129 is composite, 127 is the largest prime divisible by 16383 . The sum of 127 \textquotesingle s digits is $$ 1+2+7=\text { (C) } 10 $$ 