The reduction of digits in a series of two or greater digit integers
by addition to a single digit yields a sum equal to the reduction of
the digits in the sum.  For example, the series 136, 47, and 653 has a
sum of 836. The sum of these digits is 17 (8+3+6) and the sum of these
is 8 (1+7).  Now adding the digits in the series yields 10 (1+3+6); 11
(4+7); and 14 (6+5+3).  Reducing each of this to a single integer
yeilds 1 (1+0); 2 (1+1); and 6 (1+4), and the sum of these is 8
(1+2+6). So the reduced sum of the series is equal to the sum of the
reduced integers in the series. This is true regardless of the number
of integers in the series and the size of the integers (as long as
they are at least two digit).  My question is "is this property of
numbers known by any name or theorm?" How can I explain this result in
a "scientific" way.

Hi chetxx:

Thanks for the interesting question! I always love answering math questions. 

The property/phenomenon of integer numbers you are describing is often
referred to as "casting 9s" or "casting out 9s". By doing the
recursive adding of digits as you describe (or in modified, but
equivalent ways), you can easily check the validity not only of
additions of large integers, but multiplications and divisions too!

The following page talks about the general idea of casting nines (pg.
3) and then how that can be applied to checking calculations (pg. 4):

Marvelous Math Trick
URL: http://www.susancanthony.com/pdfhandouts/Mathhandout.pdf

This website shows a very simple illustration of a simpler sum:

MathPractice Instructions
URL: http://www.jumpingminds.com/MathPractice/instructions.htm
Note: Scroll to bottom of the page.

This next site gives a more "formal" treatment to the whole phenomenon:

Casting Out Nines 
URL: http://mathworld.wolfram.com/CastingOutNines.html

Some more resources as well:

Casting Out Nines
URL: http://www.jimloy.com/number/nines.htm

Math Chat: Milk-Bottle Pressure And Adding Hymns 
URL: http://www.csmonitor.com/cgi-bin/getasciiarchive?script/98/02/03/020398.feat.scitech.4

Search Strategy (on Google):
* "sum of digits"
* "sum of digits" recursive 
* "cast 9" sum of digits 
* "casting out 9s" digits
* "casting out nines" digits
* "casting out nines" digits "checking addition"

I hope this helps!

websearcher

Thanks for the quick and perfect answer to my question.  I understand!!!