What is “Casting Out Nines”? Part 3

 Learn the magic behind this amazing mathematical tool.

Jason Marshall, PhD
5-minute read
Episode #84

Why Do We Use Nine?

So we’ve now seen that if we divide an addition problem by any number we like, then the sum of the remainders will always equal the remainder of the sum. If we choose to use the number 9 as the number we divide the addition problem by, this interesting little property we’ve discovered is actually what we’ve come to know as casting out nines. But why 9? And why haven’t we been finding remainders when we’ve been doing casting out nines? Because, as it turns out, if you divide any number by 9 you can always find the remainder simply by adding up the digits of the number. In other words, finding the check digit of a number is equivalent to finding its remainder when dividing by 9. Pretty amazing, right? Go ahead and test it for yourself with some numbers and you’ll see that it always works.

So, as you can see, the casting out nines technique of checking that the sum of a group of check digits is equal to the check digit of the sum really is the same thing as checking that the sum of a group of remainders is equal to the remainder of the sum—that’s the magic behind casting out nines!

Number of the Week

Before we finish up for today, it’s time for this week’s featured number, conversion, or tip from my post on QDT’s blog The Quick and Dirty. This week’s tip will help you cast out nines even faster. It’s easiest to see how it works with an example, so let’s say you’re finding the check digit of the number 525,829. One way is to simply add the numbers from left to right: 5+2+5+8+2+9. So that’s 5+2=7, then 7+5=12, then 12+8=20, then 20+2=22, then 22+9=31, and finally we turn this into a check digit by finding that 3+1=4. 

This works, but the numbers you’re adding can start to get big. So the trick is to know that you don’t actually have to add all the numbers before finding a check digit—you can find check digits along the way to keep the numbers small, like this: 5+2=7, then 7+5=12 which has a check digit of 3, then we’re back to 3+8=11 which has a check digit of 2, then 2+2=4, and finally 4+9=13 which has a check digit of 4. Of course the answer is the same either way, but the beauty of this second method is that you’re always dealing with adding small numbers—even when finding check digits of huge numbers—so it’s almost always going to be a lot faster.

Wrap Up

Okay, that’s all the math we have time for today. Remember to become a fan of the Math Dude on Facebook where you’ll find a new featured number or math puzzle posted every weekday. And if you’re on Twitter, please follow me there too. Finally, if you have math questions, feel free to send them my way via Facebook, Twitter, or by email at mathdude@quickanddirtytips.com.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

Nine image courtesy of Shutterstock


About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.