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### Sum of the Digits of Multiples of 9

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Date: 04/08/2001 at 20:10:07
From: Jennifer Springer
Subject: The number nine

Why do the digits of nine times a number add up to 9? For example:

9 x 4  = 36,   3 + 6 = 9

9 x 10 = 90,   9 + 0 = 9

And why does this not work for 9 x 11?

9 x 11 = 99,   9 + 9 = 18

Why not 9?
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Date: 04/08/2001 at 23:08:46
From: Doctor Peterson
Subject: Re: The number nine

Hi, Jennifer.

First, let me point out that this observation can be extended to
larger numbers in two ways:

(1) As the rule for identifying a number divisible by 9. See our
Divisibility Rules FAQ at:

http://mathforum.org/dr.math/faq/faq.divisibility.html

(2) In the method of "casting out nines" to determine whether any

Casting Out Nines to Check Arithmetic
http://mathforum.org/library/drmath/view/55926.html

In the simple case, we easily prove your rule algebraically. Suppose x
is any single-digit number; then

9x = (10-1)x

= 10x - x

This almost looks like the expanded form of a decimal number, except
that the digits would have to be x and -x, and we can't have negative
digits. So we can "borrow," adding 10 to the units digit and taking 1
from the tens digit:

9x = 10x - x

= 10(x-1) + (10-x)

This tells us that the tens digit is x-1, and the units digit is 10-x.
In your example with x = 4, these are 4-1 = 3 and 10-x = 6 (making the
number 36).

But what is the sum of the digits?

(x-1) + (10-x) = 9

Now, why doesn't this work if x > 10? Because then x-1 and 10-x would
not both be single digits, so there would be additional carries or
borrows needed, which would mess up the trick a bit - but not
completely. The sum of the digits will still be a MULTIPLE of 9 - and
that's the trick for recognizing whether a number is divisible by 9.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
Middle School Algebra