## Casting Out the 9′s

August 9, 2008 by Liberius

*Lesson 1.1, Part 4*

In Part 2, we discussed how to compute Digit Sums. If you did the practice problems (I hope you all did!), you should see how straightforward the process is. This is good and all, but what about when you want to find the Digit Sum of a numbers such as say…97637254. This could end up being quite tedious (in case you were wondering the Digit Sum is 7), but in today’s lesson I’ll show you how you can compute it quite easily!

One of the last things we learned in Part 3 (The Nine-Point Circle) was that adding ’9′ to a digit sum was equivalent to adding ’0′ to the digit sum…neither ended up changing the final Digit Sum. For example, 13 and 139 have the same Digit Sum.

13 —-> (1 + 3) —-> 4

139 —-> (1 + 3 + 9) —-> 13 —-> (1 + 3) —-> 4

Ultimately what this means is that when computing Digit Sums, you can throw-out/ignore any 9′s you see. This also includes any numbers that add up to ’9′. For example, in the number 154, you can ignore the ’5′ & ’4′ since “5 + 4 = 9″.

So 154 becomes **1**~~54~~ (which means the Digit Sum is ’1′). If you don’t believe me…

154 —-> (1 + 5 + 4) —-> 10 —-> (1 + 0) —-> 1

Now look at that number I gave you in the beginning, the one that looked like it would be tedious to compute the Digit Sum….97637254. After cancelling out the ’9′s this number becomes ~~9~~**7**~~637254~~, or just ’7′ for the Digit Sum.

This works because each time you add ’9′ to the Digit Sum you are doing one full-revolution on the Nine-Point Circle, bringing you back to the same starting point.

**Practice Problems**

Compute the following Digit Sums:

**1.** 237

**2.** 72892

**3.** 871362

**4.** 37226

**5.** 9438541

**6.** 1438273

**7.** 847382

**8.** 13956402

**9.** 57604

**10.** 848834

ANSWERS

(*Side Note:* If you end up with ’0′ as a Digit Sum, meaning everything cancelled….you can count the Digit Sum as either ’0′ or ’9′ since they both lie in the same spot on the Circle)

**Other Parts of Lesson 1.1:**

Part One – What are Digit Sums?

Part Two – Computing Digit Sums

Part Three – The Nine-Point Ciricle

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*Related*

on August 9, 2008 at 8:50 pm |LiberiusThere’s a neat little trick that you can use to help solve some of these problems…especially the last one. See if you can figure it out; if you do post it below so others can see! Once you find out the technique go back and see if it helps make any of the other problems easier….

on August 9, 2008 at 8:52 pm |What are Digit Sums? « Knowledge & Wisdom[...] Parts of Lesson 1.1 Part Two – Computing Digit Sums Part Three – The Nine-Point Ciricle Part Four – Casting Out the 9’s Possibly related posts: (automatically generated)Puzzle – quick sumSimple Divisibility [...]

on August 9, 2008 at 8:52 pm |Computing Digit Sums « Knowledge & Wisdom[...] Other Parts of Lesson 1.1: Part One – What are Digit Sums? Part Three – The Nine-Point Circle Part Four – Casting Out the 9’s Possibly related posts: (automatically generated)Series differences, and fascinating factsPuzzle – [...]

on August 9, 2008 at 8:53 pm |The Nine-Point Circle « Knowledge & Wisdom[...] Other Parts of Lesson 1.1: Part One – What are Digit Sums? Part Two – Computing Digit Sums Part Four – Casting Out the 9’s [...]