> summing up their digits recursively gives 3 6 or 9

What does this part mean? For example 57.

57 is 3 times 19.

The standard divisibility rule for 3, 6 and 9 in base 10 is to sum the digits until you only have one left and check if it's one of those. Here, 5+7=12, 1+2=3, so 57 is divisible by 3.

5 plus 7 is 12, which of course has a 1 and a 2, the sum of which is 3.