Take an arbitrary number - for the sake of an example, I'll use 392. If we add the digits, we get 3 + 9 + 2 = 14, and then add those digits to get 5 (keep adding the digits of each result until it reduces to a single-digit number).
Compare that to this method: let q and r be the quotient and remainder of the number with respect to 10. Take q+r, and repeat with that new result until it reduces to a single digit number. So, for 392, we do 39+2 = 41, and 4+1 = 5.
In this case, the two methods end up with the same result. I haven't been able to find any counter examples. Is this guaranteed to happen?