One of the features of base 12 (also known as duodecimal or dozenal) is that it generally has nicer divisibility patterns than other bases (although in terms of divisibility rules alone, base 6 reigns supreme).

I will use these digits: 0 1 2 3 4 5 6 7 8 9 * # (the last two digits are called dek and el).

1
This is here for completeness; every integer is divisible by 1.
2
The number ends in 0, 2, 4, 6, 8, or *.
3
The number ends in 0, 3, 6, or 9.
4
The number ends in 0, 4, or 8.
5
Start with the rightmost pair of digits, subtract the next pair of digits to the left, add the next pair of digits, subtract the next, etc. If the resulting number is divisible by 5, the whole number is. Simple example: 3136 is divisible by 5, because 36 - 31 = 5. So is 7#2#. If borrowing threatens to make a subtraction more trouble than it's worth, add 5 or * first.
6
The number ends in 0 or 6.
7
Add and subtract triples of digits similarly to the rule for base 5. If the result is divisible by 7, the whole number is.
8
The number ends with an even digit followed by 0 or 8, or an odd digit followed by 4.
9
The last two digits of the number form a number which is divisible by 9.
*
The number is divisible by 5 and 2.
#
The sum of the digits is divisible by #.

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