Not really sure to what extent these are useful, but here they are. Probably better to look them up as needed for specific problems, apart from some easy ones like 9, 3, 4
Divisibility by 2
A number is divisible by 2 if the number is even. This means the final digit of the number ends with a (0, 2, 4, 6, or 8)
Divisibility by 3
A number is divisible by 3 if the sum of the number's digits is divisible by 3
Divisibility by 4
A number is divisible by 4 if the final two digits of the number form a number that is divisible by 4
Divisibility by 5
A number is divisible by 5 if the final digit is a 0 or a 5
Divisibility by 6
A number is divisible by 6 if the number is divisible by both 2 and 3
Divisibility by 7
We start by removing the final digit from the number, double the number and subtract it away from the shortened original number
If the number formed is a 0 or a number that is divisible by 7, the original number is divisible by 7. We can repeat the process as needed
Divisibility by 8
A number is divisible by 8 if the final three digits of the number form a number that is divisible by 8
Divisibility by 9
A number is divisible by 9 if the sum of the number's digits is divisible by 9
Divisibility by 10
A number is divisible by 10 if the final digit is a 0
Divisibility by 11
Form the sum of the digits in the odd places and subtract away the sum of the digits in the even places
The odd places refer to the 1st digit, 3rd digit, 5th digit, and so on
The even places refer to the 2nd digit, 4th digit, 6th digit, and so on
Note, we can count our digits starting from right to left or left to right. This can result in a sign difference, which is not important.
If the result is a 0 or a number that is divisible by 11, then the original number is divisible by 11
Divisibility by 12
A number is divisible by 12 if the number is divisible by both 3 and 4