SOLUTION: can you show me proof on the divisibility theories of both 9 and 11?

Question 5425: can you show me proof on the divisibility theories of both 9 and 11?
Answer by mrduffner(11) (Show Source): You can put this solution on YOUR website!
Let me start by saying that the tricks people use to find whether a number is divisible by 1-13 are just that, tricks. They are not "theories." My teacher mentioned them when I was in school, and we had to copy them down, and I thought, "Wow, this is great, I'll use these all of the time." But believe me, you won't. When you're sitting there solving an equation, you're not going to be thinking "Hmm, well if I add all of the digits together and divide by three..." or "Ok, the sum of all of the even digits, subtracted from the sum of all of the odd digits... divided by 11... 0 counts..." Believe me, you won't. You'll be using a calculator. But if you can remember all of the tricks, and want to do them in your head every time you see a big number, I envy you. It never hurts to do a little mental math, and that is probably my weakest point. Anyway, for your reference, here are those tricks.

The trick for finding out if a number is divisible by 9 is:
Add all of the digits of the number together. If the sum is divisible by 9, the number is also. Example: 189: 1+8+9=18, 18/9=2, 189/9=21. 153: 1+5+3=9, 9/9=1, 153/9=17.

The trick for finding out if a number is divisible by 11 is different. I don't fully understand it myself. I believe that in a three digit number at least, you subtact the number in the ones place from the two numbers in the hundreds and tens place. I'm probably confusing you. Let's try 275: 27-5=22, 22/11=2, 275/11=25. Here's another one- 187: 18-7=11, 11/11=1, 187/11=17. What I don't understand is, what happens in numbers >999?

I hope this helps. Good luck! :)
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