Problem:
Convert $8687$ to base $35$.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Convert the following to base }35: \\
 8687_{10} \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Determine }\text{the }\text{powers }\text{of }\text{35 }\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-35 }\text{representation }\text{of }8687: \\
 
\begin{array}{|c|c|c}
\hline
 \text{Power} & \text{Base}^{\text{Power}} & \text{Place value} \\
\hline
 3 & 35^3 & 42875 \\
\hline
 2 & 35^2 & 1225 \\
 1 & 35^1 & 35 \\
 0 & 35^0 & 1 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Label }\text{each }\text{place }\text{of }\text{the }\text{base-35 }\text{representation }\text{of }\text{8687 }\text{with }\text{the }\text{appropriate }\text{power }\text{of }35: \\
 
\begin{array}{ccccccc}
 \text{Place} &   &   & 35^2 & 35^1 & 35^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  &   \\
 8687_{10} & = & ( & \_\_ & \_\_ & \_\_ & )_{\text{}_{35}} \\
\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{8687 }\text{in }\text{base }35: \\
 
\begin{array}{l}
 \frac{8687}{35}=248 \text{with remainder} \fbox{$7$} \\
 
\begin{array}{ccccccc}
 \text{Place} &   &   & 35^2 & 35^1 & 35^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  &   \\
 8687_{10} & = & ( & \_\_ & \_\_ & 7 & )_{\text{}_{35}} \\
\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{8687 }\text{in }\text{base }35: \\
 
\begin{array}{l}
 \frac{248}{35}=7 \text{with remainder} \fbox{$3$} \\
 
\begin{array}{ccccccc}
 \text{Place} &   &   & 35^2 & 35^1 & 35^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  &   \\
 8687_{10} & = & ( & \_\_ & 3 & 7 & )_{\text{}_{35}} \\
\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{8687 }\text{in }\text{base }35: \\
 
\begin{array}{l}
 \frac{7}{35}=0 \text{with remainder} \fbox{$7$} \\
 
\begin{array}{ccccccc}
 \text{Place} &   &   & 35^2 & 35^1 & 35^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  &   \\
 8687_{10} & = & ( & 7 & 3 & 7 & )_{\text{}_{35}} \\
\end{array}
 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{number }8687_{10} \text{is }\text{equivalent }\text{to }737_{35} \text{in }\text{base }35. \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & 8687_{10}=737_{35} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/number_theory/convert_base_w_steps/1520.txt'}