The Magic of Number 9 (Part 1)

Have you ever noticed that the number 9 is kind of amazing? What's that…did I hear you say "NO?" Then prepare yourself to be amazed and keep on reading to learn all about the magic behind the mysterious number 9.

Jason Marshall, PhD,
December 7, 2013
Episode #177

Page 2 of 2

How to Get a Multiple of 9

Subtracting the digit sum of a number from itself always gives a multiple of 9.

The first step is to find what's called the digit sum—which is just a fancy way of saying the number you get when you add up all the digits of the original number. We're then told to subtract this digit sum from the original number.

The question to think about is: What happens when we do this? To see, let's imagine we've started with some 2-digit number which we'll write as xy—where x is the number in the 10s place and y is the number in the 1s place. For example, x would be 7 and y would be 5 for the number 75.

The first thing to notice here is that the number xy can also be written 10x + y. The next thing to notice is that the digit sum of the number xy is just x + y. Which—and here comes the sneaky part—means that subtracting the digit sum of a number from that number always gives (10x + y) - (x + y) = 10x + y - x - y = 9x.

Big deal? Yes, this is actually a huge deal. It says that subtracting the digit sum of a number from itself always gives a multiple of 9. And this turns out to be key for both of our puzzles.

A Common Thread

We've now seen that the first few steps of Natalie's puzzle have turned our number into a multiple of 9. And if we look back at Cynthias "orange kangaroo" puzzle, we can also see that the first few steps there were also designed to have us end up with a number that's a multiple of 9 (since multiplying any number by 9—as we were instructed to do—always gives a multiple of 9).

And that, my friends, is precisely where the paths of these two very different looking puzzles cross. Because, believe it or not, now that we've figured out that both puzzles start by turning our random number into a multiple of that amazing number 9, we're just a short hop away from discovering out how both puzzles work.

But, unfortunately, we're all out of time for today. Which means that the exciting conclusion will have to wait until next time.

Wrap Up

Okay, that's all the math we have time for today.

Please be sure to check out my book The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.

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!

Number 9 image from Shutterstock.


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