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What Is "Casting Out Nines"? Part 2

Learn more about using the “casting out nines” method to quickly and easily check your answers to multiplication and division problems.

By
Jason Marshall, PhD
Episode #083

We’ve recently talked about several quick and dirty methods that you can use to make sure your answers to arithmetic problems have the correct signs and parities, and we also talked about an amazing trick called “casting out nines” that you can use to check whether or not your answers to addition problems are correct. But, believe it or not, this is just the tip of the iceberg because, as we’ll see today, casting out nines can be used to check a lot more than just addition. In fact, it can be used to check your answers to subtraction, multiplication, and division problems, too.

Review: How to Use Casting Out Nines to Check Addition

Since everything we’re going to learn today relies upon the techniques we talked about in Part 1 of this series,let’s take a minute to review how to use casting out nines to check your answers to addition problems. All you really need to know is how to turn any number into a check digit. To do that, just add the digits of the number repeatedly until you’re left with a single digit number. For example, to calculate the check digit of the number 618, start by adding the digits to get 6+1+8=15. Is that a single digit number? No, so repeat the process and add up the digits of this new number to get 1+5=6...that’s the check digit of 618.

To check your answer to an addition problem, just remember that the check digit of the answer must be the same as the check digit of the sum of the check digits of all the numbers in the problem. For example, can 63+19+22=104 be correct? Well, start by finding the check digits of the three numbers in the problem. As you can calculate, the check digits of 63, 19, and 22 are 9, 1, and 4. Next, add these to get 9+1+4=14, which has a check digit of 5. If the answer 104 is correct, its check digit must match this number. Since the check digit of 104 is indeed 5, we see that 63+19+22=104 may indeed be true.

Does Casting Out Nines Guarantee Correct Answers?

It’s important to point out that just because casting out nines has told us that the answer we calculated for this addition problem may be correct, we have no guarantee that the answer is correct. After all, there are lots of numbers besides 104 that have a check digit of 5 (such as 203 or 500). So, no, casting out nines doesn’t confirm correct answers with complete accuracy. But it does always expose wrong answers since they’ll always give different check digits that tell you it’s time to go back and try again.

How to Use Casting Out Nines to Check Multiplication

[[AdMiddle]Good news: Once you understand how to use casting out nines to check addition, you also know how to use it to check multiplication since the process is basically the same...except we’re now going to use multiplication instead of addition. For example, let’s use casting out nines to check 231 x 45 = 10,385. Can this be the correct answer?

Well, start as before by finding the check digits of the numbers in the problem: 231 has a check digit of 6 and 45 has a check digit of 9. Now, since this is a multiplication problem and not an addition problem, we need to multiply these check digits (instead of adding them as before) to get 6 x 9 = 54. Next, find that the check digit of this number is 9. If the answer 10,385 is correct, its check digit must also be 9. As you can calculate, the check digit of 10,385 is 8, which means that the answer is wrong. Time to try again.

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About the Author

Jason Marshall, PhD
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