\(A=4^{17}+4^{18}+4^{19}+4^{20}+4^{17}\left(999-4\right)\)
\(=4^{17}+4^{18}+4^{19}+4^{20}+999.4^{17}-4^{18}\)
\(=4^{17}+4^{19}+4^{20}+999.4^{17}\)
\(=4^{17}\left(1+4^2+4^3\right)+999.4^{17}\)
\(=81.4^{17}+999.4^{17}\)
\(\left\{{}\begin{matrix}81⋮9\\999⋮9\end{matrix}\right.\Rightarrow A⋮9\)