elementary number theory - Divisibility Rules for Bases other than $10$

I still remember the feeling, when I learned that a number is divisible by $3$, if the digit sum is divisible by $3$. The general way to get these rules for the regular decimal system is asked/answered here: Divisibility rules and congruences. Now I wonder, what divisibility rules an alien with $12$ (or $42$) fingers would come up with?

So let $n=\sum_k c_k b^k$ be the representation with base $b$. Looking at some examples shows indicate that $n$ is divisible by $b-1$, if $\sum_k c_k$ is. This seems to be a poor man's extension to the decimal divisibility by $9$ rule.

The answer to the above mentioned question, says that "One needn't memorize motley exotic divisibility tests. ". Motley exotic divisibility tests are very welcome here!

Answer

Claim 1:

The divisibility rule for a number '$a$' to be divided by '$n$' is as follows. Express the number '$a$' in base '$n+1$'. Let '$s$' denote the sum of digits of '$a$' expressed in base '$n+1$'. Now $n|a \iff n|s$. More generally, $a \equiv s \pmod{n}$.

Example:

Before setting to prove this, we will see an example of this. Say we want to check if $13|611$. Express $611$ in base $14$.

$$611 = 3 \times 14^2 + 1 \times 14^1 + 9 \times 14^0 = (319)_{14}$$
where $(319)_{14}$ denotes that the decimal number $611$ expressed in base $14$. The sum of the digits $s = 3 + 1 + 9 = 13$. Clearly, $13|13$. Hence, $13|611$, which is indeed true since $611 = 13 \times 47$.

Proof:

The proof for this claim writes itself out. Let $a = (a_ma_{m-1} \ldots a_0)_{n+1}$, where $a_i$ are the digits of '$a$' in the base '$n+1$'.

$$a = a_m \times (n+1)^m + a_{m-1} \times (n+1)^{m-1} + \cdots + a_0$$
Now, note that
$$\begin{align} n+1 & \equiv 1 \pmod n\\ (n+1)^k & \equiv 1 \pmod n \\ a_k \times (n+1)^k & \equiv a_k \pmod n \end{align}$$
$$\begin{align} a & = a_m \times (n+1)^m + a_{m-1} \times (n+1)^{m-1} + \cdots + a_0 \\ & \equiv (a_m + a_{m-1} \cdots + a_0) \pmod n\\ a & \equiv s \pmod n \end{align}$$
Hence proved.

Claim 2:
The divisibility rule for a number '$a$' to be divided by '$n$' is as follows. Express the number '$a$' in base '$n-1$'. Let '$s$' denote the alternating sum of digits of '$a$' expressed in base '$n-1$' i.e. if $a = (a_ma_{m-1} \ldots a_0)_{n-1}$, $s = a_0 - a_1 + a_2 - \cdots + (-1)^{m-1}a_{m-1} + (-1)^m a_m$. Now $n|a$ if and only $n|s$. More generally, $a \equiv s \pmod{n}$.

Example:

Before setting to prove this, we will see an example of this. Say we want to check if $13|611$. Express $611$ in base $12$.

$$611 = 4 \times 12^2 + 2 \times 12^1 + B \times 12^0 = (42B)_{12}$$
where $(42B)_{14}$ denotes that the decimal number $611$ expressed in base $12$, $A$ stands for the tenth digit and $B$ stands for the eleventh digit.
The alternating sum of the digits $s = B_{12} - 2 + 4 = 13$. Clearly, $13|13$. Hence, $13|611$, which is indeed true since $611 = 13 \times 47$.

Proof:

The proof for this claim writes itself out just like the one above. Let $a = (a_ma_{m-1} \ldots a_0)_{n+1}$, where $a_i$ are the digits of '$a$' in the base '$n-1$'.

$$a = a_m \times (n-1)^m + a_{m-1} \times (n-1)^{m-1} + \cdots + a_0$$
Now, note that
$$\begin{align} n-1 & \equiv (-1) \pmod n\\ (n-1)^k & \equiv (-1)^k \pmod n \\ a_k \times (n-1)^k & \equiv (-1)^k a_k \pmod n \end{align}$$

$$\begin{align} a & = a_m \times (n-1)^m + a_{m-1} \times (n-1)^{m-1} + \cdots + a_0 \\ & \equiv ((-1)^m a_m + (-1)^{m-1} a_{m-1} \cdots + a_0) \pmod n\\ a & \equiv s \pmod n \end{align}$$
Hence proved.

Pros and Cons:

The one obvious advantage of the above divisibility rules is that it is a generalized divisibility rule that can be applied for any '$n$'.

However, the major disadvantage in these divisibility rules is that if a number is given in decimal system we need to first express the number in a different base. Expressing it in base $n-1$ or $n+1$ may turn out to be more expensive. (We might as well try direct division by $n$ instead of this procedure!).
However, if the number given is already expressed in base $n+1$ or $n-1$, then checking for divisibility becomes a trivial issue.