Rules for Divisibility

Rule for Divisibility by 2:

A number will be divisible by 2 if its one’s digit is either an even number or 0.

For example: 28, 30, 486, 357914 are divisible by 2.

Rule for Divisibility by 3: 

A number will be divisible by 3 if the sum of the digits of the number is divisible by 3.

For example:

492 :  4 + 9 + 2 = 15 ; 15 is divisible by 3, so 492 is also divisible by 3.

450028: 4 + 5 + 0 + 0 + 2 + 8 = 19; 19 is not divisible by 3, so 450028 is not divisible by 3.

Rule for Divisibility by 4: 

A number will be divisible by 4 if its last two digits are divisible by 4. The number which have two or more than two zeros at the end of the number is also divisible by 4.

For example:

7000:  more than two zeros, so 7000 is divisible by 4.

85628 : 28 is divisible by 4, so 85628 is divisible by 4.

Rule for Divisibility by 5: 

A number will be divisible by 5 if it ends with 5 or 0.

For example:

390, 2675, 28000 are divisible by 5.

Rule for Divisibility by 6: 

A number will be divisible by 6 if it is divisible by both 2 and 3.

For example:

69342: its one’s digit is 2 so it is divisible by 2 and sum of its digits( 6 + 9 + 3 + 4 + 2) is 24 which is divisible by 3, so it is also divisible by 3. It satisfies both the conditions therefore it is divisible by 6.

154: its one digit is an even number so it is divisible by 2 but the sum of its digits (1 + 5 + 4) is 10 which is not divisible by 3. Since it satisfies only one condition, so it is not divisible by 6.

Rule for Divisibility by 7:

For checking divisibility by 7 we do following steps:

Example 1)  112: 11 2 : multiply the one’s digit by 2 and subtract the result by the remaining number, if the result after subtraction is divisible by 7 then the number is divisible by 7.

11 – 2 ⅹ 2 = 11- 4 = 7 ; 7 is divisible by 7, so 112 is divisible by 7.

Example 2) 2961: 296 1 :  296 – 1ⅹ2 = 296 – 2 = 294
Now repeat the same steps for 294: 29 4 : 29 – 4 ⅹ 2 = 29 – 8 = 21, 21 is divisible by 7

Therefore 2961 is divisible by 7.

Rule for Divisibility by 8:

A number will be divisible by 8 if the last three digits of a number is divisible by 8. Also, if the last three digits of a number are zeros, the number is divisible by 8.

For example:

980120: 120 is divisible by 8, so 980120 is also divisible by 8.

76543000: the last three digits are 0, so 76543000 is also divisible by 8.

Rule for Divisibility by 9:

A number will be divisible by 9 if the sum of the digits of the number is divisible by 9.

For example:

68931: 6 + 8 + 9 + 3 + 1 = 27, 27 is divisible by 9, so 68931 is also divisible by 9.

74390454: 7 + 4 + 3 + 9 + 0 + 4 + 5 + 4= 36, 36 is divisible by 9, so 74390454 is also divisible by 9.

Divisibility by 10:

Any number which ends with zero is divisible by 10.

For example:

786590, 230, 60 are divisible by 10.

Divisibility by 11:

A number will be divisible by 11 if the sum of digits at odd and even places are equal or differ by a number divisible by 11.

Example 1)  601216:

Sum_odd = 6 + 1 + 1 = 8, Sum_even = 0 + 2 + 6 = 8

Sum_odd = Sum_even , so 601216 is divisible by 11.

Example 2)  8241673:

Sum_odd = 8 + 4 + 6 + 3 = 21, Sum_even =  2 + 1 + 7 = 10

Sum_odd – Sum_even = 21 – 10 = 11, so 8241673 is divisible by 11.

               
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