Problem:
Convert $9295$ to base $30$.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Convert the following to base }30: \\
 9295_{10} \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Determine }\text{the }\text{powers }\text{of }\text{30 }\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-30 }\text{representation }\text{of }9295: \\
 
\begin{array}{|c|c|c}
\hline
 \text{Power} & \text{Base}^{\text{Power}} & \text{Place value} \\
\hline
 3 & 30^3 & 27000 \\
\hline
 2 & 30^2 & 900 \\
 1 & 30^1 & 30 \\
 0 & 30^0 & 1 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Label }\text{each }\text{place }\text{of }\text{the }\text{base-30 }\text{representation }\text{of }\text{9295 }\text{with }\text{the }\text{appropriate }\text{power }\text{of }30: \\
 
\begin{array}{ccccccc}
 \text{Place} &   &   & 30^2 & 30^1 & 30^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  &   \\
 9295_{10} & = & ( & \_\_ & \_\_ & \_\_ & )_{\text{}_{30}} \\
\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{9295 }\text{in }\text{base }30: \\
 
\begin{array}{l}
 \frac{9295}{30}=309 \text{with remainder} \fbox{$25$} \\
 
\begin{array}{ccccccc}
 \text{Place} &   &   & 30^2 & 30^1 & 30^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  &   \\
 9295_{10} & = & ( & \_\_ & \_\_ & 25 & )_{\text{}_{30}} \\
\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{9295 }\text{in }\text{base }30: \\
 
\begin{array}{l}
 \frac{309}{30}=10 \text{with remainder} \fbox{$9$} \\
 
\begin{array}{ccccccc}
 \text{Place} &   &   & 30^2 & 30^1 & 30^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  &   \\
 9295_{10} & = & ( & \_\_ & 9 & 25 & )_{\text{}_{30}} \\
\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{9295 }\text{in }\text{base }30: \\
 
\begin{array}{l}
 \frac{10}{30}=0 \text{with remainder} \fbox{$10$} \\
 
\begin{array}{ccccccc}
 \text{Place} &   &   & 30^2 & 30^1 & 30^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  &   \\
 9295_{10} & = & ( & 10 & 9 & 25 & )_{\text{}_{30}} \\
\end{array}
 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Convert }\text{all }\text{digits }\text{that }\text{are }\text{greater }\text{than }\text{9 }\text{to }\text{their }\text{base }\text{30 }\text{alphabetical }\text{equivalent}: \\
 
\begin{array}{ccc}
 \text{Base-30 digit value} &   & \text{Base-30 digit} \\
 10 & \rightarrow  & \text{a} \\
 9 & \rightarrow  & 9 \\
 25 & \rightarrow  & \text{p} \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{number }9295_{10} \text{is }\text{equivalent }\text{to }\text{a9p}_{30} \text{in }\text{base }30. \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & 9295_{10}=\text{a9p}_{30} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/number_theory/convert_base_w_steps/3042.txt'}