Answer
The quotient of the given division in base four is, \[{{20}_{\text{four}}}\].
Work Step by Step
The division in base four can be performed similar to that in base ten.
First, divide the first two digits of the dividend,\[{{10}_{\text{four}}}\]by \[{{2}_{\text{four}}}\]. In the given table, the largest product, in the vertical column of 2, that is less than or equal to \[{{10}_{\text{four}}}\] is \[{{10}_{\text{four}}}\].
Since, \[{{2}_{\text{four}}}\times {{2}_{\text{four}}}={{10}_{\text{four}}}\], the first number in the quotient is \[{{2}_{\text{four}}}\]:
\[{{2}_{\text{four}}}\overset{2}{\overline{\left){{{100}_{\text{four}}}}\right.}}\]
Now, perform the multiplication \[{{2}_{\text{four}}}\times {{2}_{\text{four}}}={{10}_{\text{four}}}\] and write \[{{10}_{\text{four}}}\]under the first two digits of the dividend:
\[{{2}_{\text{four}}}\overset{2}{\overline{\left){\begin{align}
& {{100}_{\text{four}}} \\
& \underline{10} \\
\end{align}}\right.}}\]
Now, perform the subtraction:
\[{{10}_{\text{four}}}-{{10}_{\text{four}}}=0\]
\[{{2}_{\text{four}}}\overset{2}{\overline{\left){\begin{align}
& {{100}_{\text{four}}} \\
& \underline{10} \\
\end{align}}\right.}}\]
0
Now, drop down the next digit in the dividend, \[0\]:
\[{{2}_{\text{four}}}\overset{2}{\overline{\left){\begin{align}
& {{100}_{\text{four}}} \\
& \underline{10} \\
\end{align}}\right.}}\]
00
Since,0 is less than 2, write a 0 after 2 in the quotient:
\[{{2}_{\text{four}}}\overset{{{20}_{\text{four}}}}{\overline{\left){\begin{align}
& {{100}_{\text{four}}} \\
& \underline{10} \\
\end{align}}\right.}}\]
00
Thus, the obtained quotient is \[{{20}_{\text{four}}}\].
Now, to check whether the above obtained solution is correct, perform the division by converting thedivisor, the dividend and the quotient into base ten:
\[{{2}_{\text{four}}}=2\], \[{{100}_{\text{four}}}=16\] and \[{{20}_{\text{four}}}=8\]
Since,\[2\overset{8}{\overline{\left){16}\right.}}\] is indeed true, the answer obtained is correct.
Hence, the quotient of the given division in base four is, \[{{20}_{\text{four}}}\].