Digital Root Divisibility Test Not Working?

Hello everyone! I'm new here and this is my first post. I came here to get a better grasp on my mathematics so I can learn programming better. And with that:

Here is my question: I've been using the digital root test for find out the divisibility for numbers, and it has always worked, but now that I've learned prime factorization, it doesn't seem to work so well for larger numbers(or even for small numbers for that matter). When I attempt it on a larger number, I often still get a remainder when I try to divide it w/ the digital root and the number actually is not divisible by the digital root.

Here are a couple examples:

The number 37 ---> turns out to be divisible by 10? Should this be a 1? Also, would that make this number an automatic composite number(if 10) or prime(if 1)?

347 ----> Turns out to be divisible by 5 using the digital root testing method, but when you actually try to divide 347 by 5, you get 69.4!

I suppose what I'm asking is, what are the rules concerning digital root divisibility testing? Why does it not work all of the time on numbers?