Divisibility Rules

Divisibility Rules Of 2,3,4,5,6,7,8,9,10,11,12,13

1.Divisibility Rule of 2 : 

A number is divisible by 2 if the number is even. All even number have the right most digit as 0,2,4,6,8. Therefore one just have to inspect the rightmost digit of the number and if it is ending in 0,2,4,6,8 then the number is divisible by 2.

Example:  the numbers 2540, 5682, 254664, 5896, 5458 are all divisible by 2.


2.Divisibility Rule of 3:  If the sum of digits of a number is divisible by 3, then the number itself is divisible
   by 3..
Example: If we have to check for the number 255.

                     First add all the digits of 255 
                     we get,  2+5+5 = 12.
 And we know that 12 is divisible by 3. Hence the number 255 is divisible by3.


3. Divisibility Rule of 4: To check divisibility by 4, We pick the last two digits of the number and if they are divisible by 4, then the entire number is divisible by 4.

Example:Lets say we have to check for 2542524. 
This is a large number and if we have to divide this number by 4 then it will take a bit of time. Instead we will just pick the last two digits,here its is 24 and just check whether 24 is divided by 4 exactly.

Now. 24/4 = 4. Therefore the number 2542524 is divisible by 4.


4. Divisibility Rule of 5: Any number that ends with either 0 or 5 is divisible by 5.

Example: 21500, 2147985, 43546576785 are all divisible by 5.


5. Divisibility Rule of 6: The number 6 is obtained by multiplying the two prime numbers 2 and 3. If the number is divisible by both 2 and 3 then the entire number will be divisible by 6.

Example:  The check whether the number 45954 is divisible by 6.
1. Check for 2 :  45954 ends with 4 and hence is even and hence divisible by 2.
2. Check for 3: The sum of the digits of 45954 is 4+5+9+5+4 = 27 And we know 27 is divisible by 3 and hence the number 45954 is divisible by3

We have checked that the number 45954 is divisible by both 2 and 3. Hence the number 45954 is divisible by 6.


6. Divisibility Rule of 7: To check for the number being divisible by 7, we first
    1.  take the last digit and double it.
    2. Subtract the number obtained in step1 from the number that remains after the taking the last digit apart.

Example: to check whether 343 is divisible by 7 or not.
        1. Double the last digit. Here it is 3. 3x2= 6.
        2. Now subtract 6 from the remaining number, that is , 34-6 = 28.
    And we know that 28 is a multiple of 7. Hence the number 343 is divided by 7.

Note: If the number is large then we can go on repeating the above 2 steps till we get a manageable number.


7.  Divisibility Rule of 8:  To check divisibility by 8, We pick the last three digits of the number and if they are divisible by , then the entire number is divisible by 8.

Example : To check whether 164684647895320 is divisible by 8 or not we just have to check the last 3 digits of this number. They are 320 and 320/8 = 40. Hence 320 is divided by 8 and hence the entire number 164684647895320 is divisible by 8.


8. Divisibility Rule of 9:  Rule for 9 is same as rule for 3 except the sum of digits obtained should be divided by 9.

Example: The sum of digits of 725481 is 7+2+5+4+8+1 = 27 which is divisible by 9. Hence the number 725481 is divisible by 9.


9. Divisibility Rule of 10: Rule of 10 is just to check the last digit of a number. If it is zero then the entire number is divisible by 0.

Example: 10, 46830, 4156746870, 5440 are all divisible by 10.


10.Divisibility Rule of 11: For this we have to subtract the sum of digits at odd places from the sum of digits at even places. If the result is either 0 or a number that is divisible by 11, then the entire number itself is divisible by 11.

For example: If we have to check 3901414 for divisibility by 11 then we add the digits at the odd places  4+4+0+3  to get 11 and the sum of digits at even places to get  1+1+9 = 11
.
Now, If we subtract them we will get 11-11 = 0 and hence the number 3901414 is divisible by 11.


11.Divisibility Rule of 12: A number is divisible by 12 if it is divisible by 3 and 4 both.

Example: to check for 546216

1. Check for divisibility by 3: Add the digits 5+4+6+2+1+6 = 24 and 24/3 = 6. Hence divisible by 3.


2. Check for divisibility by 4: The last two digits are 16 and they are divisible by 4. Hence the number 546216 is divisible by 4.

546216 is divisible by 3 and 4 both, therefore the number 544516 is divisible by 12.