We all know that in base-10 the digits of multiples of 3 sum multiples of 3 and that any number in base-10 the digits of which sum multiples of 3 will be divisible by 3. Same with 9. It occurred to me that this is a general truth for all bases. Could be stated something like:
For any positive integers b and n such that n divides b-1 without remainder, the digits of multiples of n in base-b notation will sum a number divisible by n without remainder, and all numbers in base-b notation the digits of which sum a multiple of n will be divisible by n without remainder.
So in base-13, twelve, six, four, three, and two all behave like 3 and 9 in base-10. As does 1, but that's a trivial case. What is this theorem called in its general form? (As opposed to just the arithmetic trick for 3 and 9) Bonus: does said theorem have any applicability in searching for primes? I'm thinking something like using a base one higher than some superior highly composite number, like base-13, so you can quickly check for divisibility into many other numbers with the "3 and 9" trick.
