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Mathematics, Numbers, and Computers - How Is "Casting Out Nines" Used To Check The Results Of Addition Or Multiplication?

How is "casting out nines" used to check the results of addition or multiplication?

"Casting out nines" is a method of checking the accuracy of an answer in an addition or multiplication problem. As illustrated in the multiplication example below, the process begins by adding the digits in both the multiplicand (one of the numbers that is being multiplied) and the multiplier (the other number being multiplied). In this example, it gives the results of "13" and "12," respectively. Since these results are greater than 9, the figures are "cast out" and the operation is repeated until the sum of the digits is less than 9. Repeated calculations here give the results as "4" and "3," respectively.

The next step is to multiply the resulting "excess" from the multiplicand by the "excess" from the multiplier (4 x 3 below). Add the digits of the result until they yield a number equal to or less than 9, as...

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