Divisibility Rules

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Dividing by 2
All even numbers are divisible by 2. E.g., all numbers ending in 0,2,4,6 or 8.

Dividing by 3
Add up all the digits in the number.
Find out what the sum is. If the sum is divisible by 3, so is the number
For example: 12123 (1+2+1+2+3=9) 9 is divisible by 3, therefore 12123 is too!


Dividing by 4
Are the last two digits in your number divisible by 4?
If so, the number is too!
For example: 358912 ends in 12 which is divisible by 4, thus so is 358912.

Dividing by 5
Numbers ending in a 5 or a 0 are always divisible by 5.

Dividing by 6
If the Number is divisible by 2 and 3 it is divisible by 6 also.

Dividing by 7 (2 Tests)
Take the last digit in a number.
Double and subtract the last digit in your number from the rest of the digits.
Repeat the process for larger numbers.
Example: 357 (Double the 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7 and we can now say that 357 is divisible by 7.

NEXT TEST
Take the number and multiply each digit beginning on the right hand side (ones) by 1, 3, 2, 6, 4, 5. Repeat this sequence as necessary
Add the products.
If the sum is divisible by 7 - so is your number.
Example: Is 2016 divisible by 7?
6(1) + 1(3) + 0(2) + 2(6) = 21
21 is divisible by 7 and we can now say that 2016 is also divisible by 7.

Dividing by 8
This one's not as easy, if the last 3 digits are divisible by 8, so is the entire number.
Example: 6008 - The last 3 digits are divisible by 8, therefore, so is 6008.

Dividing by 9
Almost the same rule and dividing by 3. Add up all the digits in the number.
Find out what the sum is. If the sum is divisible by 9, so is the number.
For example: 43785 (4+3+7+8+5=27) 27 is divisible by 9, therefore 43785 is too!

Dividing by 10
If the number ends in a 0, it is divisible by 10.

Dividing by 11
The difference between the sum of the odd numbered digits (1st, 3rd, 5th...) and the sum of the even numbered digits (2nd, 4th...) is divisible by 11.

Divisibility by 11 is the most interesting of the above tests . We do two sums (the odd numbered digits and the even numbered digits), subtract one sum from the other, and see if this is divisible by 11. By the way, if we end up with zero, then that is divisible by 11. We can repeat that process, just as we did with 3. Let's look at an example:

34871903
3+8+1+0=12
4+7+9+3=23
23-12=11 Is divisible by 11
We can, of course, do the summing in different orders. In fact we can just go from left to right adding and subtracting alternate digits: 3-4+8-7+1-9+0-3=-11 (divisible by 11).

Note: For the time being, I will not deal with divisibility by larger numbers. Since they are hardly useful in exams like CAT, GMAT, GRE, SNAP and etc.