Problem:
Convert $7294$ to base $13$.
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Convert the following to base }13: \\
 7294_{10} \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \text{Determine }\text{the }\text{powers }\text{of }\text{13 }\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-13 }\text{representation }\text{of }7294: \\
 
\begin{array}{|c|c|c}
\hline
 \text{Power} & \text{Base}^{\text{Power}} & \text{Place value} \\
\hline
 4 & 13^4 & 28561 \\
\hline
 3 & 13^3 & 2197 \\
 2 & 13^2 & 169 \\
 1 & 13^1 & 13 \\
 0 & 13^0 & 1 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{Label }\text{each }\text{place }\text{of }\text{the }\text{base-13 }\text{representation }\text{of }\text{7294 }\text{with }\text{the }\text{appropriate }\text{power }\text{of }13: \\
 
\begin{array}{cccccccc}
 \text{Place} &   &   & 13^3 & 13^2 & 13^1 & 13^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 7294_{10} & = & ( & \_\_ & \_\_ & \_\_ & \_\_ & )_{\text{}_{13}} \\
\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{7294 }\text{in }\text{base }13: \\
 
\begin{array}{l}
 \frac{7294}{13}=561 \text{with remainder} \fbox{$1$} \\
 
\begin{array}{cccccccc}
 \text{Place} &   &   & 13^3 & 13^2 & 13^1 & 13^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 7294_{10} & = & ( & \_\_ & \_\_ & \_\_ & 1 & )_{\text{}_{13}} \\
\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{7294 }\text{in }\text{base }13: \\
 
\begin{array}{l}
 \frac{561}{13}=43 \text{with remainder} \fbox{$2$} \\
 
\begin{array}{cccccccc}
 \text{Place} &   &   & 13^3 & 13^2 & 13^1 & 13^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 7294_{10} & = & ( & \_\_ & \_\_ & 2 & 1 & )_{\text{}_{13}} \\
\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{7294 }\text{in }\text{base }13: \\
 
\begin{array}{l}
 \frac{43}{13}=3 \text{with remainder} \fbox{$4$} \\
 
\begin{array}{cccccccc}
 \text{Place} &   &   & 13^3 & 13^2 & 13^1 & 13^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 7294_{10} & = & ( & \_\_ & 4 & 2 & 1 & )_{\text{}_{13}} \\
\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{7294 }\text{in }\text{base }13: \\
 
\begin{array}{l}
 \frac{3}{13}=0 \text{with remainder} \fbox{$3$} \\
 
\begin{array}{cccccccc}
 \text{Place} &   &   & 13^3 & 13^2 & 13^1 & 13^0 &   \\
   &   &   & \downarrow  & \downarrow  & \downarrow  & \downarrow  &   \\
 7294_{10} & = & ( & 3 & 4 & 2 & 1 & )_{\text{}_{13}} \\
\end{array}
 \\
\end{array}
 \\
\end{array}
 \\
 
\begin{array}{l}
 \text{The }\text{number }7294_{10} \text{is }\text{equivalent }\text{to }3421_{13} \text{in }\text{base }13. \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & 7294_{10}=3421_{13} \\
\end{array}
$} \\
\end{array}
 \\
\end{array}

{'source': 'AMPS', 'file': 'amps/mathematica/number_theory/convert_base_w_steps/917.txt'}