# Divisibility rules using sum of digits

1. Sep 4, 2010

### ershi

I saw someone discussing divisibility rules in another thread and would thought I would make a note that the divisibility rule of 9 of summing the digits to see if you end up with 9 is really a trick of the counting base you are using (base 10).
In general, this divisibility rule applies to all bases but with different numbers in each base.
Whatever the last symbol of the base is, if the digits sum to that number in that base, it is divisible by the integer represented by the last symbol.
For instance,
in base 13, the number that works is 12,
AND
the trick also works for all of these numbers, as they are factors of 12:
6, 4, 3, 2

Adding digits in different bases can feel tricky at first because they have to be interpreted in the same base as you are using.
For instance if you come across "B5" (B in base 13 = 11 in base 10)
you should turn this into "13",
and then summing this becomes 4

this tells you B5(a base 13 representation) is divisible by 4

2. Sep 5, 2010

### sachinism

you are correct but we mostly work with numbers involving base 10 only in number theory or to say most day to day mathematics

3. Dec 7, 2010

### twokicks

I found the related rule which can be shown to be true:

for b, a which are positive integers:
b divides the sum of the digits of a in base b^2 + 1
iff
b divides a.

btw, does anyone know any good references for this subject? I found one or two semi-related papers via wikipedia and google scholar, but I was looking for like a textbook reference.