How do you do the casting out nines?

It was a very common rule used in the primary school. During that time there weren´t any calculators in the classroom to check the right result of a certain mathematical operation. It is known as “the casting out of nines” or “the rule of the nine out”. It can be used to validate the result of the four basic arithmetic operations: the addition, the subtraction, the multiplication and the division.

I prefer to give you a small example explaining “how” rather than give you the theory that explains “why”. There are several techniques that can be used for this checking to become faster. I won’t use these techniques. I will just try to explain how to use it as easy as possible.

Let’s check the following addition: 9787 + 4526 = 14,313

• The first step is to repeatedly add up all the digits of 9787: 9 + 7 + 8 + 7 = 31 = 3 + 1 = 4
• The second step is to repeatedly add up the digits of 4526: 4 + 5 + 2 + 6 = 17 = 1 + 7 = 8
• The third step is to repeatedly add up the digits of 14,313: 1 + 4 + 3 + 1 + 3 = 12 = 1 + 2 = 3
• Finally we check if the addition of the first with the second is equal to the third: 4 + 8 = 12 = 1 + 2 = 3

Let’s check the following multiplication: 9974 x 2348 = 23,418,952

• The first step is to repeatedly add up all the digits of 9974: 9 + 9 + 7 + 4 = 29 = 2 + 9 = 11 = 1 + 1 = 2
• The second step is to repeatedly add up all the digits of 2348: 2 + 3 + 4 + 8 = 17 = 1 + 7 = 8
• The third step is to repeatedly add up all the digits of 23.418.952: 2 + 3 + 4 + 1 + 8 + 9 + 5 + 2 = 34 = 3 + 4 = 7
• Finally we check if the multiplication of the first with the second is equal to the third: 2 x 8 = 16 = 1 + 6 = 7

Attention: this rule is used to prove that a certain operation is wrong, but it is not used to prove it is right!

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