Music and Math Project

Multiplication Table

We can also multiply two numbers in the Land of Twelve. Then, we first need to figure out what are the multiples of small numbers. In other words, we need to find out what is the table of multiplication in the Land of Twelve. This is the result:
\(\times\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(\mathrm{X}\) \(\mathrm{Y}\) \(10\)
\(1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(\mathrm{X}\) \(\mathrm{Y}\) \(10\)
\(2\) \(2\) \(4\) \(6\) \(8\) \(\mathrm{X}\) \(10\) \(12\) \(14\) \(16\) \(18\) \(1\mathrm{X}\) \(20\)
\(3\) \(3\) \(6\) \(9\) \(10\) \(13\) \(16\) \(19\) \(20\) \(23\) \(26\) \(29\) \(30\)
\(4\) \(4\) \(8\) \(10\) \(14\) \(18\) \(20\) \(24\) \(28\) \(30\) \(34\) \(38\) \(40\)
\(5\) \(5\) \(\mathrm{X}\) \(13\) \(18\) \(21\) \(26\) \(2\mathrm{Y}\) \(34\) \(39\) \(42\) \(47\) \(50\)
\(6\) \(6\) \(10\) \(16\) \(20\) \(26\) \(30\) \(36\) \(40\) \(46\) \(50\) \(56\) \(60\)
\(7\) \(7\) \(12\) \(19\) \(24\) \(2\mathrm{Y}\) \(36\) \(41\) \(48\) \(53\) \(5\mathrm{X}\) \(65\) \(70\)
\(8\) \(8\) \(14\) \(20\) \(28\) \(34\) \(40\) \(48\) \(54\) \(60\) \(68\) \(74\) \(80\)
\(9\) \(9\) \(16\) \(23\) \(30\) \(39\) \(46\) \(53\) \(60\) \(69\) \(76\) \(83\) \(90\)
\(\mathrm{X}\) \(\mathrm{X}\) \(18\) \(26\) \(34\) \(42\) \(50\) \(5\mathrm{X}\) \(68\) \(76\) \(84\) \(92\) \(\mathrm{X}0\)
\(\mathrm{Y}\) \(\mathrm{Y}\) \(1\mathrm{X}\) \(29\) \(38\) \(47\) \(56\) \(65\) \(74\) \(83\) \(92\) \(\mathrm{X1}\) \(\mathrm{Y}0\)
\(10\) \(10\) \(20\) \(30\) \(40\) \(50\) \(60\) \(70\) \(80\) \(90\) \(\mathrm{X}0\) \(\mathrm{Y}0\) \(100\)

Long multiplication

Now we know the multiplication table, we can calculate the product of two big numbers for instance by using long multiplication \[ \begin{array}{cccc} 4 & 0 & 9 &\\ 2 & 5 & \mathrm{X} & \times \\ \hline & & ? \end{array} \] First we do the multiplication by \(\mathrm{X}\). After calculating the multiplication table, we know that \(\mathrm{X} \times 9 = 76 \). We write the last digit, the \(6\), on the last line, and put the \( 7 \) on the position for the multiples of twelve above the top row. We get \[ \begin{array}{cccc} & 7 & \\ 4 & 0 & 9 &\\ 2 & 5 & \mathrm{X} & \times \\ \hline & & 6 \end{array} \] Then, we multiply \(0 \times \mathrm{X} = 0\). We add the \(7\) that we needed to remember, and get \(7\) on the last row. \[ \begin{array}{cccc} & & \\ 4 & 0 & 9 &\\ 2 & 5 & \mathrm{X} & \times \\ \hline & 7 & 6 \end{array} \] Next, \(\mathrm{X} \times 4 = 34 \), so \[ \begin{array}{ccccc} & & & & \\ & 4 & 0 & 9 &\\ & 2 & 5 & \mathrm{X} & \times \\ \hline 3 & 4 & 7 & 6 \end{array} \] We now need to do the multiplications by \(5\). Remember that the \(5\) stands for \(5\) multiples of twelve. So we first have to add a \(0\) on the new line \[ \begin{array}{ccccc} & & & & \\ & 4 & 0 & 9 &\\ & 2 & 5 & \mathrm{X} & \times \\ \hline 3 & 4 & 7 & 6\\ & & ? & 0 \end{array} \] Now, we will keep going in the same spirit \[ \begin{array}{cccccc} & & & & & \\ & & 4 & 0 & 9 &\\ & & 2 & 5 & \mathrm{X} & \times \\ \hline & 3 & 4 & 7 & 6\\ 1 & 8 & 3 & 9 & 0 \end{array} \] \[ \begin{array}{cccccc} & & & & & \\ & & 4 & 0 & 9 &\\ & & 2 & 5 & \mathrm{X} & \times \\ \hline & 3 & 4 & 7 & 6\\ 1 & 8 & 3 & 9 & 0\\ 8 & 1 & 6 & 0 & 0 \end{array} \] Finally we add the last lines \[ \begin{array}{cccccc} & & & & & \\ & & 4 & 0 & 9 &\\ & & 2 & 5 & \mathrm{X} & \times \\ \hline & 3 & 4 & 7 & 6\\ 1 & 8 & 3 & 9 & 0\\ 8 & 1 & 6 & 0 & 0 & + \\ \hline \mathrm{X} & 1 & 2 & 4 &6 & \end{array} \]

Exercises

Do the following multiplications:
  1. \(4\mathrm{X} \times 5\)
  2. \(\mathrm{Y}9 \times 46\)
  3. \(63 \times 77\)
  4. \(51 \times 82\)
  5. \(66 \times \mathrm{Y}3 \)
  6. \(86 \times 29\)
  7. \(740 \times \mathrm{X}48\)
  8. \(577 \times 346\)
  9. \(462 \times \mathrm{YX}8\)
  10. \(923 \times 203\)