3. The sum of the digits of the number $9^{2014}$ is $a$. The sum of the digits of $a$ is $b$, and the sum of the digits of $b$ is $c$. Determine the number $c$.
Solution. From 9|9014 it follows that $9 \mid a$. From $9^{2014}<10^{2014}$ it follows that $a<2015 \cdot 9=18135$. From 9|a and $a<18135$ it follows that $9 \mid b$ and $b<5 \cdot 9=45$. From $9 \mid b$ and $b<45$ it follows that $9 \mid c$ and $c<2 \cdot 9=18$. Finally, $c=9$.