The Rule of Nine

This procedure of repeatedly adding the digits of a number until we get to a single digit is known as calculating the digit root [en.wikipedia.org]. There are several interesting tricks to calculate the digit roots and digit roots are put to use in a few interesting ways (one of which the OP linked to). The Wikipedia page I've linked to describes some of these.

Here's an interesting trick that the Wikipedia page does not describe: when calculating the digit root, you can always remove 9s and if you end up with a 0 in this way, the digit root is 9. So if you encounter a 9 at any time, you can simply remove them; the digit root of 8597 is the same as that of 857! Also, any time the sum exceeds 9, you can simply subtract 9 and proceed!

With the above trick, this is how you'll proceed to calculate the digit sum of 8597:

Code:

8597 --> 857 (removing 9)
8 + 5 = 13
13 - 9 = 4 (removing 9)
4 + 7 = 11
11 - 9 = 2 (removing 9)

The proof for why this trick works should be relatively simple.