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# The Magic of Number 9 (Part 2)

The number 9 is kind of amazing. How amazing? Keep on reading for the exciting conclusion to Math Dude's story about the magical number 9.

By
Jason Marshall, PhD,
December 14, 2013
Episode #178

Page 2 of 2

## How to Prove It

Let's say we have a 2-digit multiple of 9 number that we can write xy. The x here stands for the digit in the 10s place and the stands for the digit in the 1s place. If you think about it, you'll see that we can also write this number 10x + y (since that's just what the decimal number system means). Now here's the clever part: Let's rewrite this number as 10xy = (9xx) + y = 9x + (x + y)—in other words, a multiple of 9 plus the sum of the original digits.

If you think about this, you'll see that it says that the sum of the digits is equal to our original multiple of 9 number minus some other multiple of 9 number. The fact that both of these are multiples of 9 must mean that the sum of the digits of a multiple of 9 number must also be a multiple of 9. And it means that the digital root of a multiple of 9 number must be a multiple of 9, too!

Of course, we only showed this for a 2-digit number, but you can use the same technique to show that it's also true for numbers with however many digits you like. It's a bit more work to do, but it does work!

## Orange Kangaroos in Denmark

So what does any of this have to do with those orange kangaroos in Denmark? It's actually surprisingly simple!

Once you know that the math part of the puzzle was designed solely to make us come up with the number 9, and then to subtract 5 from this number to arrive at the number 4. In other words, no matter what number you start with, the amazing properties of the number 9 that we've discovered guarantee that you'll always end up with the number 4…and therefore with the letter "D."

No matter what number you start with…you'll always end up with the number 4.

Then, since Denmark is the only country in Europe that starts with "D" in English, you're pretty much guaranteed to choose it. And once you've arrived at Denmark, kangaroo and orange are the most common things that people come up with—so they make for a fairly safe bet.

The tricky part is that all of the math at the beginning seems sufficiently complicated to ensure that everybody who randomly picks a number must end up at a different letter of the alphabet. Which makes knowing you'd come up with Denmark seem quite remarkable! But the truth is that all of that math was there to actually guarantee that it would happen! Pretty clever, right?

## 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 courtesy of Shutterstock.