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Caveman's multiplication


Today, I am going to tell you a method of doing multiplication by the ancients. I do not remember the name of this method, nor do I know when this method was invented. Let us just call it "the cave-man's multiplication method"!

In this method, for example, if we want to calculate $345 \times 23$, we will draw a rectangle grid with $2$ rows and $3$ columns. Next, we draw the diagonals of the inner grids as illustrated in the figure above. Now, we write the number $345$ above the rectangle, make sure that one digit is lying over one column. We write the number $23$ to the right of the rectangle, again each digit is for each row.

After writing the two numbers $345$ and $23$, the next step is to do the multiplication. We multiply each number of the row with each number of the column. Each inner grid is divided into two parts, the upper part is for writing the ten-digit of the result.
So we have $$2 \times 3 = 6, ~~2 \times 4 = 8, ~~2 \times 5 = 10,$$ $$3 \times 3 = 9, ~~3 \times 4 = 12, ~~3 \times 5 = 15.$$

Now, to get the result, we add all the numbers up based on the diagonals.
In the first diagonal, we have $5$.

In the second diagonal we have $0 + 1 + 2 = 3$.

Next, $1 + 8 + 1 + 9 = 19$, we write $9$ down and the carry $1$ is for the next diagonal.

Next, $6 +$ carry $1 = 7$.

Finally we have the result $345 \times 23 = 7935$!









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