Answer
The multiplication of two given numbers in base six is, \[{{4443}_{\text{six}}}\].
Work Step by Step
Since, the computation involves base six, the only digits which are allowed are 0, 1, 2, 3, 4 and 5.
The procedure to multiply two numbers in base sixis same as in base ten.
\[{{543}_{\text{six}}}\]
\[\underline{\times {{5}_{\text{six}}}}\]
Hence, first multiply \[{{5}_{\text{six}}}\] with \[{{3}_{\text{six}}}\], which is above it in first column:
\[{{5}_{\text{six}}}\times {{3}_{\text{six}}}={{15}_{\text{ten}}}=\left( 2\times 6 \right)+\left( 3\times 1 \right)={{23}_{\text{six}}}\]
Now, write 3 in the first column below the horizontal line and carry 2 to the second column:
\[5\overset{2}{\mathop{4}}\,{{3}_{\text{six}}}\]
\[\underline{\times {{5}_{\text{six}}}}\]
\[{{3}_{\text{six}}}\]
Now, multiply \[{{5}_{\text{six}}}\] with \[{{4}_{\text{six}}}\], which is in the second column, and add \[{{2}_{\text{six}}}\] to the product:
\[\left( {{5}_{\text{six}}}\times {{4}_{\text{six}}} \right)+{{2}_{\text{six}}}=20+2={{22}_{\text{ten}}}=\left( 3\times 6 \right)+\left( 4\times 1 \right)={{34}_{\text{six}}}\]
Now, write 4 in the second column below the horizontal line and carry 3 to the third column:
\[\overset{3}{\mathop{5}}\,{{43}_{\text{six}}}\]
\[\underline{\times {{5}_{\text{six}}}}\]
\[{{43}_{\text{six}}}\]
Now, multiply \[{{5}_{\text{six}}}\] with \[{{5}_{\text{six}}}\], which is in the third column, and add \[{{3}_{\text{six}}}\] to the product:
\[\left( {{5}_{\text{six}}}\times {{5}_{\text{six}}} \right)+{{3}_{\text{six}}}=25+3={{28}_{\text{ten}}}=\left( 4\times 6 \right)+\left( 4\times 1 \right)={{44}_{\text{six}}}\]
Write \[{{44}_{\text{six}}}\] in front of \[{{43}_{\text{six}}}\] below the horizontal line:
\[{{543}_{\text{six}}}\]
\[\underline{\times {{5}_{\text{six}}}}\]
\[{{4443}_{\text{six}}}\]
Now, to check whether the above obtained solution is correct, perform the multiplication by converting each number to base ten:
\[{{543}_{\text{six}}}=207\], \[{{5}_{\text{six}}}=5\] and \[{{4443}_{\text{six}}}=1035\]
Since, \[207\times 5\] indeed equals 1035, the solution obtained is correct.
Hence, the multiplication of two given numbers in base six is, \[{{4443}_{\text{six}}}\].
