Problem:
Convert $\text{4gc}_{25}$ to base 9.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Convert the following to base }9: \\
 \text{4gc}_{25} \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{First, }\text{convert }\text{4gc}_{25} \text{to }\text{base }10: \\
 \text{Find }\text{the }\text{base-10 }\text{representation }\text{of }\text{4gc}_{25} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Split }\text{4gc}_{25} \text{into }\text{its }\text{component }\text{digits}: \\
 
\begin{array}{c}
 \text{Base-25 digit} \\
 4 \\
 \text{g} \\
 \text{c} \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Convert }\text{all }\text{alphabetical }\text{digits }\text{to }\text{their }\text{numerical }\text{equivalents}: \\
 
\begin{array}{ccc}
 \text{Base-25 digit} &   & \text{Base-25 digit value} \\
 4 & \rightarrow  & 4 \\
 \text{g} & \rightarrow  & 16 \\
 \text{c} & \rightarrow  & 12 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Label }\text{each }\text{digit }\text{of }\text{4gc}_{25} \text{with }\text{its }\text{place}: \\
 
\begin{array}{cc}
 \text{Base-25 digit value} & \text{Place} \\
 4 & 25^2 \\
 16 & 25^1 \\
 12 & 25^0 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Write }\text{the }\text{product }\text{that }\text{each }\text{base-25 }\text{digit }\text{represents }\text{in }\text{base }10: \\
 
\begin{array}{cccc}
 \text{Base-25 digit value} & \text{Place} & \rightarrow  & \text{Base-10 equivalent} \\
 4 & 25^2 & \rightarrow  & 4\ 25^2 \\
 16 & 25^1 & \rightarrow  & 16\ 25^1 \\
 12 & 25^0 & \rightarrow  & 12\ 25^0 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Compute }\text{the }\text{base-10 }\text{equivalent }\text{of }\text{4gc}_{25}: \\
 
\begin{array}{cccccc}
\hline
 \text{Base-25 digit value} & \text{Place} & \rightarrow  & \text{Base-10 equivalent} &   & \text{Value} \\
\hline
 4 & 25^2 & \rightarrow  & 4\ 25^2 &   & 2500 \\
\hline
 16 & 25^1 & \rightarrow  & 16\ 25^1 & + & 400 \\
\hline
 12 & 25^0 & \rightarrow  & 12\ 25^0 & + & 12 \\
\hline
   &   &   &   & = & 2912 \\
\hline
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{number }\text{4gc}_{25} \text{is }\text{equivalent }\text{to }2912_{10} \text{in }\text{base }10. \\
 \text{4gc}_{25}=2912_{10} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Now }\text{convert }\text{2912 }\text{to }\text{base }9: \\
 \text{Now, }\text{find }\text{the }\text{base-}9 \text{representation }\text{of }2912 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Determine }\text{the }\text{powers }\text{of }9 \text{that }\text{will }\text{be }\text{used }\text{as }\text{the }\text{places }\text{of }\text{the }\text{digits }\text{in }\text{the }\text{base-}9 \text{representation }\text{of }2912: \\
 
\begin{array}{|c|c|c}
\hline
 \text{Power} & \text{Base}^{\text{Power}} & \text{Place value} \\
\hline
 4 & 9^4 & 6561 \\
\hline
 3 & 9^3 & 729 \\
 2 & 9^2 & 81 \\
 1 & 9^1 & 9 \\
 0 & 9^0 & 1 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Label }\text{each }\text{place }\text{of }\text{the }\text{base-}9 \text{representation }\text{of }\text{2912 }\text{with }\text{the }\text{appropriate }\text{power }\text{of }9: \\
 
\begin{array}{cccccccc}
 \text{Place} &   &   & 9^3 & 9^2 & 9^1 & 9^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 2912_{10} & = & ( & \_\_ & \_\_ & \_\_ & \_\_ & )_{\text{}_9} \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Determine }\text{the }\text{value }\text{of }\text{the }\text{first }\text{digit }\text{from }\text{the }\text{right }\text{of }\text{2912 }\text{in }\text{base }9: \\
 
\begin{array}{l}
 \frac{2912}{9}=323 \text{with remainder} \fbox{$5$} \\
 
\begin{array}{cccccccc}
 \text{Place} &   &   & 9^3 & 9^2 & 9^1 & 9^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 2912_{10} & = & ( & \_\_ & \_\_ & \_\_ & 5 & )_{\text{}_9} \\
\end{array}
 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Determine }\text{the }\text{value }\text{of }\text{the }\text{next }\text{digit }\text{from }\text{the }\text{right }\text{of }\text{2912 }\text{in }\text{base }9: \\
 
\begin{array}{l}
 \frac{323}{9}=35 \text{with remainder} \fbox{$8$} \\
 
\begin{array}{cccccccc}
 \text{Place} &   &   & 9^3 & 9^2 & 9^1 & 9^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 2912_{10} & = & ( & \_\_ & \_\_ & 8 & 5 & )_{\text{}_9} \\
\end{array}
 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Determine }\text{the }\text{value }\text{of }\text{the }\text{next }\text{digit }\text{from }\text{the }\text{right }\text{of }\text{2912 }\text{in }\text{base }9: \\
 
\begin{array}{l}
 \frac{35}{9}=3 \text{with remainder} \fbox{$8$} \\
 
\begin{array}{cccccccc}
 \text{Place} &   &   & 9^3 & 9^2 & 9^1 & 9^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 2912_{10} & = & ( & \_\_ & 8 & 8 & 5 & )_{\text{}_9} \\
\end{array}
 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Determine }\text{the }\text{value }\text{of }\text{the }\text{last }\text{remaining }\text{digit }\text{of }\text{2912 }\text{in }\text{base }9: \\
 
\begin{array}{l}
 \frac{3}{9}=0 \text{with remainder} \fbox{$3$} \\
 
\begin{array}{cccccccc}
 \text{Place} &   &   & 9^3 & 9^2 & 9^1 & 9^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 2912_{10} & = & ( & 3 & 8 & 8 & 5 & )_{\text{}_9} \\
\end{array}
 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{number }2912_{10}, \text{and }\text{therefore }\text{the }\text{number }\text{4gc}_{25}, \text{is }\text{equivalent }\text{to }3885_9 \text{in }\text{base }9. \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & \text{4gc}_{25}=2912_{10}=3885_9 \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/number_theory/convert_base_w_steps/5659.txt'}