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# Testing answers by casting out 9’s

This test is based uon a remarkable property of 9. When any number is divided by 9, the remainder will equal the sum of the number’s digits, or the sum of the number’s digits after “casting out” 9’s. This may sound strange but the idea itself if easy to understand, as the following examples will show.

**Example 1:** 16 / 9 = 1, with 7 remainder. The digit sum of 16 (1 + 6) is also 7, the same as the remainder after dividing 16 by 9.

**Example 2:** 24 / 9 = 2, with 6 remainder. The digit sum of 24 (2 + 4) is also 6.

**Example 3:** 38 / 9 = 4, with 2 remainder. Of course, the digit sum of 38 is not 2 by 11. However, after casting out 9, such as by subracting 9 from 11, the remainder deos become 2, the same as the remainder after dividing 38 by 9.

Remainders obtained by casting out 9’s can be used to provide “check numbers” for testing your answers in addition, subtraction, multiplication, and division.

You can use any of the four methods to cast out 9’s and obtain the remainders, as shown in the following examples. The same numbers are used in each example to demonstrate that the remainders obtained are the same for all four methods.

**Casting out 9’s by dividing**

Divide the number by 9 to obtain the remainder:

- Numbers – Remainder

46 – 1

85 – 4

198 – 0

6,368 – 5

**Casting out 9’s by adding**

Add the number’s digits, and add the digits in the sum if there is more than one digit in the sum, to obtain a one-digit remainder. If the remainder is 9 or a number evenly divisible by 9 (18, 27, 36, etc), count the remainder as zero.

- Number – Remainder

46: 4 + 6 = 10; 1 + 0 = 1

85: 8 + 5 = 13; 1 + 3 = 4

198: 1 + 9 + 8 = 18; 1 + 8 = 9; 0

6,368: 6 + 3 + 6 + 8 = 23; 2 + 3 = 5

**Casting out 9’s by subtracting**

Add the number’s digits and subtract 9 each time the sum equals or exceeds 9. Do this until the final remainder becomes less than 9. Or, you can first get the digit sum and then keep subtracting 9 until the remainder becoems less than 9.

- Number – Remainder

46: 4 + 6 = 10; -9 = 1

85: 8 + 5 = 13; -9 = 4

198: 1 + 9 = 10; -9 = 1; + 8 = 9; -9 = 0

also, 198: 1 + 9 + 8 = 18; -9 = 9; -9 = 0

6,368: 6 + 3 = 9; -9 = 0; + 6 + 8 = 14; -9 = 5

also, 6,368: 6 + 3 + 6 + 8 = 23; -9 = 14; – 8 = 5

**Casting out 9’s by omitting them**

Add the number’s digits but do not include 9’s or combinations equal to 9 (6 and 3; 5 and 4; 1, 3, and 5, etc). Omitting 9’s and 9-combinations saves adding them up and subtracting 9’s later. If the sum comes to 9 or more, cast out 9’s to obtain the remainder.

- Number – Remainder

46: no 9 to omit; 4 + 6 = 10; -9 = 1

85: no 9 to omit; 8 + 5 = 13; -9 = 4

198: omit 1 + 8 and omit 9; = 0

6,368: omit 6 + 3; 6 + 8 = 14; -9 = 5

Did you notice that the remainders were the same for all four methods?

Omitting 9’s is the easiest way to cast them out, especially with large numbers, since it reduces the adding and subtracting, and requires no division. In a number like 2,756,942, for instance, the 2 and 7, 5 and 4, and the 9 can be omitted, leaving only 6 and 2 to be added, giving a remainder of 8. Dividing 2,756,942 by 9 will, of course, also give a remainder of 8.

This method doesn’t give positive proof that your answer is right (no method does) but if your answer does not meet the test of casting out 9’s, you can be sure that there is an error somewhere, either in doing the calculation, or in making the check, or in both.