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Sum of Digits of Multiples of Nine

Date: 08/12/2004 at 05:19:50
From: Saba
Subject: number theory: multiples of 9

Why is it that when you add the individual digits of any multiple of 
nine until a single digit answer is reached the answer is always nine?  
Is it possible to prove this?

For example, 99 => 9 + 9 = 18 => 1 + 8 = 9

Why doesn't it work with other numbers between 1-9 either?  

Date: 08/12/2004 at 10:10:23
From: Doctor Luis
Subject: Re: number theory: multiples of 9

Hi Saba,

Good job finding that pattern!  The reason is that the sum of the 
digits of ANY multiple of 9 is also a multiple of 9.  Since you keep 
adding the digits (each time getting a new multiple of 9, but a 
smaller multiple), eventually you'll end up with a single digit 
number.  Eventually you'll get to the multiple 9 itself.

Now, how do I know that the sum of the digits is always a multiple of
9?  Suppose that a number N has digits a,b,c,d,...(from right to left),

  N = a + 10b + 100c + 1000d + ...
    = a + (b + 9b) + (c + 99c) + (d + 999d) + ...
    = (a + b + c + d + ...) + (9b + 99c + 999d + ...)
    = (a + b + c + d + ...) + 9*(b + 11c + 111d + ...)
  N = (sum of digits of N) + 9 * (some number)

Now, look at that equation carefully.  It means that

  (sum of digits of N) = N - 9 * (some number)

Since N is assumed to be a multiple of 9, we can write it in terms of 
another integer k, so that N = 9k

  (sum of digits of N) = 9 * k  -  9 * (some number)
                       = 9 * (k - (some number))
                       = 9 * (some other number)

Since we showed that the sum of the digits is 9 times some integer, 
then it is also a multiple of 9 itself.

To summarize, starting from a multiple of 9, you keep adding the 
digits, each time arriving to a multiple of 9.  This establishes a 
chain of decreasing multiples of 9, until eventually you reach 9 (from 
a two-digit multiple).  Does that make sense?

It doesn't work for other integers because the chain is broken.  For 
example, multiples of 8 such as 56 don't add up to a multiple of 8. 

Well, I hope this helped!

Let us know if you have any more questions.

- Doctor Luis, The Math Forum

Date: 08/12/2004 at 12:48:19
From: Saba
Subject: number theory: multiples of 9

Thanks loads for answering--your proof is very cool.  You guys are the 
Associated Topics:
High School Number Theory

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