Problem:
Convert $1382_{17}$ to base 6.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Convert the following to base }6: \\
 1382_{17} \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{First, }\text{convert }1382_{17} \text{to }\text{base }10: \\
 \text{Find }\text{the }\text{base-10 }\text{representation }\text{of }1382_{17} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Split }1382_{17} \text{into }\text{its }\text{component }\text{digits}: \\
 
\begin{array}{c}
 \text{Base-17 digit} \\
 1 \\
 3 \\
 8 \\
 2 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Label }\text{each }\text{digit }\text{of }1382_{17} \text{with }\text{its }\text{place}: \\
 
\begin{array}{cc}
 \text{Base-17 digit value} & \text{Place} \\
 1 & 17^3 \\
 3 & 17^2 \\
 8 & 17^1 \\
 2 & 17^0 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Write }\text{the }\text{product }\text{that }\text{each }\text{base-17 }\text{digit }\text{represents }\text{in }\text{base }10: \\
 
\begin{array}{cccc}
 \text{Base-17 digit value} & \text{Place} & \rightarrow  & \text{Base-10 equivalent} \\
 1 & 17^3 & \rightarrow  & 1\ 17^3 \\
 3 & 17^2 & \rightarrow  & 3\ 17^2 \\
 8 & 17^1 & \rightarrow  & 8\ 17^1 \\
 2 & 17^0 & \rightarrow  & 2\ 17^0 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Compute }\text{the }\text{base-10 }\text{equivalent }\text{of }1382_{17}: \\
 
\begin{array}{cccccc}
\hline
 \text{Base-17 digit value} & \text{Place} & \rightarrow  & \text{Base-10 equivalent} &   & \text{Value} \\
\hline
 1 & 17^3 & \rightarrow  & 1\ 17^3 &   & 4913 \\
\hline
 3 & 17^2 & \rightarrow  & 3\ 17^2 & + & 867 \\
\hline
 8 & 17^1 & \rightarrow  & 8\ 17^1 & + & 136 \\
\hline
 2 & 17^0 & \rightarrow  & 2\ 17^0 & + & 2 \\
\hline
   &   &   &   & = & 5918 \\
\hline
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{number }1382_{17} \text{is }\text{equivalent }\text{to }5918_{10} \text{in }\text{base }10. \\
 1382_{17}=5918_{10} \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Now }\text{convert }\text{5918 }\text{to }\text{base }6: \\
 \text{Now, }\text{find }\text{the }\text{base-}6 \text{representation }\text{of }5918 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Determine }\text{the }\text{powers }\text{of }6 \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-}6 \text{representation }\text{of }5918: \\
 
\begin{array}{|c|c|c}
\hline
 \text{Power} & \text{Base}^{\text{Power}} & \text{Place value} \\
\hline
 5 & 6^5 & 7776 \\
\hline
 4 & 6^4 & 1296 \\
 3 & 6^3 & 216 \\
 2 & 6^2 & 36 \\
 1 & 6^1 & 6 \\
 0 & 6^0 & 1 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Label }\text{each }\text{place }\text{of }\text{the }\text{base-}6 \text{representation }\text{of }\text{5918 }\text{with }\text{the }\text{appropriate }\text{power }\text{of }6: \\
 
\begin{array}{ccccccccc}
 \text{Place} &   &   & 6^4 & 6^3 & 6^2 & 6^1 & 6^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 5918_{10} & = & ( & \_\_ & \_\_ & \_\_ & \_\_ & \_\_ & )_{\text{}_6} \\
\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{5918 }\text{in }\text{base }6: \\
 
\begin{array}{l}
 \frac{5918}{6}=986 \text{with remainder} \fbox{$2$} \\
 
\begin{array}{ccccccccc}
 \text{Place} &   &   & 6^4 & 6^3 & 6^2 & 6^1 & 6^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 5918_{10} & = & ( & \_\_ & \_\_ & \_\_ & \_\_ & 2 & )_{\text{}_6} \\
\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{5918 }\text{in }\text{base }6: \\
 
\begin{array}{l}
 \frac{986}{6}=164 \text{with remainder} \fbox{$2$} \\
 
\begin{array}{ccccccccc}
 \text{Place} &   &   & 6^4 & 6^3 & 6^2 & 6^1 & 6^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 5918_{10} & = & ( & \_\_ & \_\_ & \_\_ & 2 & 2 & )_{\text{}_6} \\
\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{5918 }\text{in }\text{base }6: \\
 
\begin{array}{l}
 \frac{164}{6}=27 \text{with remainder} \fbox{$2$} \\
 
\begin{array}{ccccccccc}
 \text{Place} &   &   & 6^4 & 6^3 & 6^2 & 6^1 & 6^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 5918_{10} & = & ( & \_\_ & \_\_ & 2 & 2 & 2 & )_{\text{}_6} \\
\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{5918 }\text{in }\text{base }6: \\
 
\begin{array}{l}
 \frac{27}{6}=4 \text{with remainder} \fbox{$3$} \\
 
\begin{array}{ccccccccc}
 \text{Place} &   &   & 6^4 & 6^3 & 6^2 & 6^1 & 6^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 5918_{10} & = & ( & \_\_ & 3 & 2 & 2 & 2 & )_{\text{}_6} \\
\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{5918 }\text{in }\text{base }6: \\
 
\begin{array}{l}
 \frac{4}{6}=0 \text{with remainder} \fbox{$4$} \\
 
\begin{array}{ccccccccc}
 \text{Place} &   &   & 6^4 & 6^3 & 6^2 & 6^1 & 6^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 5918_{10} & = & ( & 4 & 3 & 2 & 2 & 2 & )_{\text{}_6} \\
\end{array}
 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{number }5918_{10}, \text{and }\text{therefore }\text{the }\text{number }1382_{17}, \text{is }\text{equivalent }\text{to }43222_6 \text{in }\text{base }6. \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & 1382_{17}=5918_{10}=43222_6 \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/number_theory/convert_base_w_steps/5579.txt'}