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To help you understand the algorithm, the video shows a simple two-digit by two-digit multiplication. You can see that the first and last digits are essentially the result of one multiplication.
The way you learned to multiply works, but computers employ a faster algorithm.
The standard method for two and three digit multiplication is delayed until fifth grade; the standard method for long division until sixth.
According to the mathematicians, from Australia and France, long multiplication used to be the only way to multiply - but proved tedious and time-consuming when the numbers became too large.