April 16th, 2012, 05:52 PM  #1 
Joined: Apr 2012 Posts: 17 Thanks: 0  Casting out 9s
I want to prove whether or not 65,547,012/26,436 = 2479 + (12,168/26,436) using casting out of 9s. I know how casting out 9s works with multiplication. However, I do not know how to apply the concept to division with a remainder/decimals. Please help me get started. Thanks! 
April 16th, 2012, 06:06 PM  #2 
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,131 Thanks: 431 Math Focus: Calculus/ODEs  Re: Casting out 9s
The dividend excess is 3. The final excess from the remaining values is . The values check.

April 16th, 2012, 06:16 PM  #3 
Joined: Apr 2012 Posts: 17 Thanks: 0  Re: Casting out 9s
Ok I see that the dividend is 6+5+5+4+7+0+1+2 has a remainder of 3 when divided by 9. I also see that 2+6+4+3+6 has a remainder of 3 and that 2+4+7+9 has a remainder of 4 and that you multiply both of those. The fraction is what is getting me. How do you know it has remainder of 0? 
April 16th, 2012, 06:19 PM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,019 Thanks: 919 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Casting out 9s
12,168 is divisible by 9 and 26,436 is divisible by 3 but not 9, so you can at least conclude that the quotient is divisible by 3.

April 16th, 2012, 06:26 PM  #5 
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,131 Thanks: 431 Math Focus: Calculus/ODEs  Re: Casting out 9s
Write your original expression as: 65,547,012 = 2479·26,436 + 12,168 then use the method: 3 = 4·3 + 0 = 3 
April 16th, 2012, 06:28 PM  #6 
Joined: Apr 2012 Posts: 17 Thanks: 0  Re: Casting out 9s
Thanks Mark. That is very clever and makes perfect sense.

April 16th, 2012, 06:37 PM  #7 
Global Moderator Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,131 Thanks: 431 Math Focus: Calculus/ODEs  Re: Casting out 9s
Glad to help! I will be honest and admit I had never heard of this method before and had to look it up. 

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