Divisibility Rules

Learning the divisibility rules is a must if you are preparing for any competition exam. It not only saves time require for calculations but is also handy at times when you opt for solving a question using options. 
Mug these divisibility rules and practise as and when you find time, you will be delighted when you apply these divisibility rules in your mocks or during your MOCK TESTS.


Divisibility by 2:- If the last digit of of a number is even ie any one amongst 0,2,4,6,8.
Divisibility by 4:- If the last 2 digits of a number are divisible by 4 or are 00, then number is divisible by 4
Divisibility by 8:- If the last 3 digits of a number are divisible by 8 or are 000, then number is divisible by 8

Example 1 :-
742 : is divisible by 2 since last digit is 2.

673 : is not divisible by 2 since last digit is 3 (which is an odd number)

67432 : is divisible by 2 as last digit is 2.
    is divisible by 4 as last 2 digits are 32 which is divisible by 4
    is divisible by 8 as last 3 digits are 432 which is divisible by 8
       

Divisibility by 3:- If sum of all the digits is divisible by 3, the number is divisible by 3.
Divisibility by 9:- If sum of all the digits is divisible by 9, the number is divivible by 9.

Example 2:-
1452 : is divisible by 3 since sum of digits is 12. but not divisible by 9 since 12 is not divible by 9.

1359 : is divisible by 3 and 9 since sum of digits is 18 which is divisible by 3 and 9.

98127 : is divisible by 3 and 9 since sum of digits is 27 which is divisible by 3 and 9.

Divisibility by 5:- If last unit's digit is 0 or 5, the number is divisible is divisible by 5.

Example 3:-
2465 :- is divisible by 5 since last digit is 5

Divisibility by 7:-
To determine if a number is divisible by 7, take the last digit off the number, double it and subtract the doubled number from the remaining number. If the result is evenly divisible by 7 (e.g. 14, 7, 0, -7, etc.), then the number is divisible by seven. This may need to be repeated several times.

Example 4:-
 476   - take off the last digit of the number which was 7
 -14   - double the removed digit and subtract it
 462   - repeat the process by taking off the 4
  -8    - and doubling it to get 8 which is subtracted
 14    - the result is 14 which is a multiple of 7

Divisibility by 11:- A number is divisible by 11, if the difference between the sum of the digits in even places and sum of the digits in the odd places is either 0 or is divisible by 11.

Example 5:-
6595149 : is divisible by 11 as difference of 6+9+1+9=25 and 5+5+4=14 is 11.
27813   :- is not divisible by 11 as sum of digits at odd places =13 and sum of digits in even places=8 their difference is neither 0 nor divisible by 11.

Divisibility by 13:- Add four times the last digit to the remaining leading truncated number. If the result is divisible by 13, then so was the first number. Apply this rule over and over again as necessary. 

Example 6:- Check if 50661 is divisible by 13
   
5066 - take last digit of number which is 1
  +4 - Multiply the number by 4 and add to truncated number.
=5070
507 - Take last digit of number which is 0
+0  - Multiply the number by 4 and add to truncated number
50  - Take last digit of number which is 7
+28 - Multiply the number by 4 and add to truncated number

this gives 78 which is 13*6. This means number is divisible by 13.

For any composite number :-
Break the number into its smallest possible coprime factors from example :-
24 = 8 x 3 :-so if the number is divisible by 8 and 3 then it is divisible by 24 as well.
12 = 4 x 3 :-so if the number is divisible by 4 and 3 then it is divisible by 12 as well.

Although 12 = 6 x 2 but this doesnot mean that every number which is divisible by 6 and 2 is divisible by 12 as well. This is because 6 and 2 are not co-primes.