This works with any integer, and it's fairly easy.

  1. Add up all the digits in the number in question
  2. Is the sum a single digit? If not, add up the digits in the answer you got to step 1.
  3. Repeat step 1 and 2 until you have a single digit for the answer.
If this digit is 9, then the number you started with is divisible (evenly) by 9 (and so are all the subsequent numbers you found).

468,373,571,595:
4+6+8+3+7+3+5+7+1+5+9+5 = 63
6+3 = 9 - number is divisible by 9.

95,361,728:
9+5+3+6+1+7+2+8 = 41
4+1 = 5 - number is not divisible by 9.

(Any number is "divisible" by 9, this writeup describes how to find ones whose quotient after being divided will be an integer. Named as such to be consistent with other nodes with a similarly formed title)