It is simply a algebra equation...
You can see it like this. Any two digit number can be multiples of our base 10 system. Such as 32 is 3*10+2.
Now since the puzzle is telling you to split up the number, they can be represented by letters. x and y; where x is 3 and y is 2.
So it now would read x*10 + y (which is the original number)
Then the puzzle tells you to add the digits together. x+y
Then to subtract that sum from the original number.
Which is
(x*10 + y) - (x+y)
Which also reads
10x - x+y-y = 9x
Therefor the answer of any two digit number will be a multiple of nine.
"My Shit is Fucked Up." --Warren Zevon and Terry Kindred.