Let $ t_{k} \left(n\right)$ show the sum of $ k^{th}$ power of digits of positive number $ n$. For which $ k$, the condition that $ t_{k} \left(n\right)$ is a multiple of 3 does not imply the condition that $ n$ is a multiple of 3?

$\textbf{(A)}\ 3  \qquad\textbf{(B)}\ 6  \qquad\textbf{(C)}\ 9  \qquad\textbf{(D)}\ 15  \qquad\textbf{(E)}\ \text{None}$