# Nines (Was Re: yuga, VarNa and colour)

jacob.baltuch at euronet.be jacob.baltuch at euronet.be
Mon May 19 16:40:56 UTC 1997

```>At 02:18 PM 5/19/97 BST, you (jacob.baltuch at euronet.be (Jacob Baltuch)) wrote:
>>>Actually the digits of any number that is a multiple of 9 will have a
>>>sum of 9 as well.
>>
>>Not true: 9 * 11 = 99
>>
>
>I think what was meant was that if you start with 99, then 9 + 9 = 18 and 1
>+ 8 = 9. I.e., supposedly all numbers that are multiples of nine will "add
>down" to nine if you make sums of their component digits till you get to a
>single-digit number.

That is true, because the sum of the digits of a number that's a multiple
of 9 will also be a multiple of 9 (not too hard to see by induction, assume
true for N = 9 * p and prove for N + 9) and (except in cases of one digit
numbers) that sum will be (strictly) smaller than the number itself.

Doing that again and again you'll have to go thru smaller and smaller
numbers which are multiples of 9 and you'll end up hitting a one digit
number which is a multiple of 9 and there aren't very many choices there.

So ok the statement is accurate if you replace "the sum of decimal digits of
a multiple of 9 is 9" by "the sum of digits of a multiple of 9 is a multiple
of 9"

```