Divisibility of integers

Generic rules for divisibility by small integers in the decimal system are well known and commonly used due to their simplicity. For instance, a number is divisible by 2 (i.e. it is even) if its last digit is divisible by 2; a number is divisible by $2^2 = 4$ if its last 2 digits are divisible by 4; and analogous rules are true for divisibility of 5 and $5^2 = 25$ as well. An integer is divisible by 3 if 3 divides the sum of its digits; this rule is also true for divisibility by 9. Similarly, an integer is divisible by 11 if 11 divides the alternating-signed sum of its digits.
In this paper, these rules are extended to integers $n$ in an arbitrary base $\beta$ .