If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is:
$\textbf{(A)}\ 5\frac{1}{3}\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 10\frac{2}{3}\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ \text{None of these answers}$
If the three numbers are in proportion to $2:4:6$, then they should also be in proportion to $1:2:3$. This implies that the three numbers can be expressed as $x$, $2x$, and $3x$. Add these values together to get: 
\[x+2x+3x=6x=64\]
Divide each side by 6 and get that 
\[x=\frac{64}{6}=\frac{32}{3}=10 \frac{2}{3}\]
which is $\boxed{\textbf{(C)}}$.