Answer
The quotient of the given division in base four is \[{{130}_{\text{four}}}\] and the remainder is \[{{1}_{\text{four}}}\].
Work Step by Step
The division in base four can be performed similar to that in base ten. First, divide the first digit of the dividend, \[{{3}_{\text{four}}}\]by \[{{2}_{\text{four}}}\]. In the given table, the largest product, in the vertical column of 2, that is less than or equal to \[{{3}_{\text{four}}}\] is \[{{2}_{\text{four}}}\]. Since, \[{{2}_{\text{four}}}\times {{1}_{\text{four}}}={{2}_{\text{four}}}\], the first number in the quotient is \[{{1}_{\text{four}}}\]:
\[{{2}_{\text{four}}}\overset{1}{\overline{\left){{{321}_{\text{four}}}}\right.}}\]
Now, perform the multiplication \[{{2}_{\text{four}}}\times {{1}_{\text{four}}}={{2}_{\text{four}}}\] and write \[{{2}_{\text{four}}}\] under the first digit of the dividend:
\[{{2}_{\text{four}}}\overset{1}{\overline{\left){\begin{align}
& {{321}_{\text{four}}} \\
& \underline{2} \\
\end{align}}\right.}}\]
Now, perform the subtraction:
\[{{3}_{\text{four}}}-{{2}_{\text{four}}}={{1}_{\text{four}}}\]
\[{{2}_{\text{four}}}\overset{1}{\overline{\left){\begin{align}
& {{321}_{\text{four}}} \\
& \underline{2} \\
\end{align}}\right.}}\]
1
Now, drop down the next digit in the dividend, 2:
\[{{2}_{\text{four}}}\overset{1}{\overline{\left){\begin{align}
& {{321}_{\text{four}}} \\
& \underline{2} \\
\end{align}}\right.}}\]
12
In the given table, the largest product, in the vertical column of 2, that is less than or equal to \[{{12}_{\text{four}}}\] is \[{{12}_{\text{four}}}\]. Since, \[{{2}_{\text{four}}}\times {{3}_{\text{four}}}={{12}_{\text{four}}}\], the second number in the quotient is \[{{3}_{\text{four}}}\]:
\[{{2}_{\text{four}}}\overset{13}{\overline{\left){\begin{align}
& {{321}_{\text{four}}} \\
& \underline{2} \\
\end{align}}\right.}}\]
12
Now, perform the multiplication \[{{2}_{\text{four}}}\times {{3}_{\text{four}}}={{12}_{\text{four}}}\] and write \[{{12}_{\text{four}}}\] under \[{{12}_{\text{four}}}\] in the above division:
\[{{2}_{\text{four}}}\overset{13}{\overline{\left){\begin{align}
& {{321}_{\text{four}}} \\
& \underline{2} \\
\end{align}}\right.}}\]
12
\[\underline{12}\]
Now, perform the subtraction:
\[{{12}_{\text{four}}}-{{12}_{\text{four}}}={{0}_{\text{four}}}\]
\[{{2}_{\text{four}}}\overset{13}{\overline{\left){\begin{align}
& {{321}_{\text{four}}} \\
& \underline{2} \\
\end{align}}\right.}}\]
12
\[\underline{12}\]
0
Now, drop down the next digit in the dividend, 1:
\[{{2}_{\text{four}}}\overset{13}{\overline{\left){\begin{align}
& {{321}_{\text{four}}} \\
& \underline{2} \\
\end{align}}\right.}}\]
12
\[\underline{12}\]
01
In the given table, the largest product, in the vertical column of 2, that is less than or equal to \[{{1}_{\text{four}}}\] is \[{{0}_{\text{four}}}\]. Since, \[{{2}_{\text{four}}}\times {{0}_{\text{four}}}={{0}_{\text{four}}}\], the third number in the quotient is \[{{0}_{\text{four}}}\]:
\[{{2}_{\text{four}}}\overset{{{130}_{\text{four}}}}{\overline{\left){\begin{align}
& {{321}_{\text{four}}} \\
& \underline{2} \\
\end{align}}\right.}}\]
12
\[\underline{12}\]
01
Now, perform the multiplication \[{{2}_{\text{four}}}\times {{0}_{\text{four}}}={{0}_{\text{four}}}\] and write \[{{0}_{\text{four}}}\] under \[{{01}_{\text{four}}}\] in the above division:
\[{{2}_{\text{four}}}\overset{{{130}_{\text{four}}}}{\overline{\left){\begin{align}
& {{321}_{\text{four}}} \\
& \underline{2} \\
\end{align}}\right.}}\]
12
\[\underline{12}\]
01
\[\underline{0}\]
Now, preform the subtraction:
\[{{1}_{\text{four}}}-{{0}_{\text{four}}}={{1}_{\text{four}}}\]
\[{{2}_{\text{four}}}\overset{{{130}_{\text{four}}}}{\overline{\left){\begin{align}
& {{321}_{\text{four}}} \\
& \underline{2} \\
\end{align}}\right.}}\]
12
\[\underline{12}\]
01
\[\underline{0}\]
1
Hence, the obtained quotient is \[{{130}_{\text{four}}}\] with the remainder \[{{1}_{\text{four}}}\].
Now, to check whether the above obtained solution is correct, perform the division by converting the divisor, the dividend and the quotient into base ten:
\[{{2}_{\text{four}}}=2\], \[{{321}_{\text{four}}}=57\] and \[{{130}_{\text{four}}}=28\]
Since, \[2\overline{\left){57}\right.}\] indeed results in quotient 28 with remainder 1, the answer obtained is correct.
Hence, the quotient of the given division in base four is \[{{130}_{\text{four}}}\] and the remainder is \[{{1}_{\text{four}}}\].
