Divisibility by 2
A number which ends with even number or 0 is always divisible by 2.
For Example 44, 120, 56, 70 etc are divisible by 2.
23, 57, 79 etc are not divisible by 2.
Divisibility by 3
For a given number, if sum of its digits is divisible by 3, then the number will be divisible by 3.
For Example: 5676 = 5 + 6 + 7 + 6 = 24, which is divisible by 3
8912 = 8 + 9 + 1 + 2 = 20, not divisible by 3
Divisibility by 4
Given number will be divisible by 4, if the last two digits are divisible by 4 and if a number with
two or more zeros at the end are also divisible by 4
For Example 4500, 134000 are divisible by 4
5228 = last two digits i.e. 28 is divisible by 4, therefore the number is divisible by 4
Divisibility by 5
Numbers ends with 5 or 0, will be divisible by 5.
For Example 465, 670, 565 will be divisible by 5.
556, 877 are not divisible by 5.
Divisibility by 6
A number divisible by 3 and 2 both will be divisible by 6. For divisibility by 3, you will use
the same rule of 3 as discussed above.
For Example 154: Divisible by 2.
1 + 5 + 4 = 10 which is not divisible by 3? Therefore, the number is not divisible by 6.
3924: Divisible by 2. Also, 3 + 9 + 2 + 4 = 18, which is divisible by 3? Therefore, the number divisible by 6
Divisibility by 7
To check divisibility by numbers like 7, 13, 17, 19, we always use one key digit known as
Osculator. Osculators for 7 is -2 i.e. negative osculator 2. Let’s see how we will use
osculator to check divisibility.
Suppose, we need to check divisibility of number 119,
The last digit of a number is to be multiplied with -2 (osculator) and then added to the remaining
part of the number. Check whether the resultant number is divisible by 7 or not.
Therefore, the number will be divisible by 7, if resultant is divisible by 7
Divisibility by 8
Given number will be divisible by 8 if the last three digits are divisible by 8 and number with
three zeros at the end are also divisible by 8
For Example 4589000, 1256 are divisible by 8
Divisibility by 9
For a given number, if sum of its digits is divisible by 9, then the number will be divisible by 9.
For Example: 5679 = 5 + 6 + 7 + 9 = 27, which is divisible by 9
8912 = 8 + 9 + 1 + 2 = 20, not divisible by 9
Divisibility by 10
A number ends with 0 is always divisible by 10
For Example 100, 2890, 4560 are divisible by 10
Divisibility by 11
If sums of digits at odd places and even places are equal or differ by a number divisible by
11, then the number will be divisible by 11
For Example 3245693
Sum of odd digits = 3 + 4 + 6 + 3 = 16
Sum of even digits = 2 + 5 + 9 = 16
Sums are equal; therefore give number is divisible by 11
Divisibility by 12
Number divisible by 4 and 3 both will be divisible by 12. For divisibility by 4 and 3, you
will use the same rule of 4 and 3 as discussed above.
For Example: 156: 1 + 5 + 6 = 12 which is divisible by 3?
Last two digits 56 are divisible by 4. Therefore, the number is divisible by 12
3924: Divisible by 4. Also, 3 + 9 + 2 + 4 = 18, which is divisible by 3? Therefore, the number
divisible by 12.
A number which ends with even number or 0 is always divisible by 2.
For Example 44, 120, 56, 70 etc are divisible by 2.
23, 57, 79 etc are not divisible by 2.
Divisibility by 3
For a given number, if sum of its digits is divisible by 3, then the number will be divisible by 3.
For Example: 5676 = 5 + 6 + 7 + 6 = 24, which is divisible by 3
8912 = 8 + 9 + 1 + 2 = 20, not divisible by 3
Divisibility by 4
Given number will be divisible by 4, if the last two digits are divisible by 4 and if a number with
two or more zeros at the end are also divisible by 4
For Example 4500, 134000 are divisible by 4
5228 = last two digits i.e. 28 is divisible by 4, therefore the number is divisible by 4
Divisibility by 5
Numbers ends with 5 or 0, will be divisible by 5.
For Example 465, 670, 565 will be divisible by 5.
556, 877 are not divisible by 5.
Divisibility by 6
A number divisible by 3 and 2 both will be divisible by 6. For divisibility by 3, you will use
the same rule of 3 as discussed above.
For Example 154: Divisible by 2.
1 + 5 + 4 = 10 which is not divisible by 3? Therefore, the number is not divisible by 6.
3924: Divisible by 2. Also, 3 + 9 + 2 + 4 = 18, which is divisible by 3? Therefore, the number divisible by 6
Divisibility by 7
To check divisibility by numbers like 7, 13, 17, 19, we always use one key digit known as
Osculator. Osculators for 7 is -2 i.e. negative osculator 2. Let’s see how we will use
osculator to check divisibility.
Suppose, we need to check divisibility of number 119,
The last digit of a number is to be multiplied with -2 (osculator) and then added to the remaining
part of the number. Check whether the resultant number is divisible by 7 or not.
Therefore, the number will be divisible by 7, if resultant is divisible by 7
Divisibility by 8
Given number will be divisible by 8 if the last three digits are divisible by 8 and number with
three zeros at the end are also divisible by 8
For Example 4589000, 1256 are divisible by 8
Divisibility by 9
For a given number, if sum of its digits is divisible by 9, then the number will be divisible by 9.
For Example: 5679 = 5 + 6 + 7 + 9 = 27, which is divisible by 9
8912 = 8 + 9 + 1 + 2 = 20, not divisible by 9
Divisibility by 10
A number ends with 0 is always divisible by 10
For Example 100, 2890, 4560 are divisible by 10
Divisibility by 11
If sums of digits at odd places and even places are equal or differ by a number divisible by
11, then the number will be divisible by 11
For Example 3245693
Sum of odd digits = 3 + 4 + 6 + 3 = 16
Sum of even digits = 2 + 5 + 9 = 16
Sums are equal; therefore give number is divisible by 11
Divisibility by 12
Number divisible by 4 and 3 both will be divisible by 12. For divisibility by 4 and 3, you
will use the same rule of 4 and 3 as discussed above.
For Example: 156: 1 + 5 + 6 = 12 which is divisible by 3?
Last two digits 56 are divisible by 4. Therefore, the number is divisible by 12
3924: Divisible by 4. Also, 3 + 9 + 2 + 4 = 18, which is divisible by 3? Therefore, the number
divisible by 12.
Divisibility by 13
Osculator for 13 = 4 The method is the same as 7 as discussed above.
See how we check the divisibility by 13, the last digit is multiplied with osculator (4), then
added to the remaining number.
Divisibility by 14
A number divisible by 7 and 2 both will be divisible by 14. To check divisibility by 7 and 2, you will use the same rules as discussed above.
Divisibility by 15
A number divisible by 5 and 3 both will be divisible by 15. Divisibility by 5 and 3 will be the same as discussed above.
Divisibility by 16
A number divisible by 8 and 2 both will be divisible by 16. Divisibility by 8 and 2 will be the same as discussed above.
Divisibility by 17
Osculator for 17 is -5 (negative osculator). Method is the same as discussed in divisibility by 7 and 13.
Divisibility by 18
Number divisible by 9 and 2, will be divisible by 18.
Divisibility by 19
Osculator for 19 is 2 (negative osculator). Method is similar as discussed in divisibility by 7 and 13.
So, Remember that
Osculator of 7 =- 2
Osculator of 13 = + 4
Osculator of 17 =- 5
Osculator of 19 = + 2
See how we check the divisibility by 13, the last digit is multiplied with osculator (4), then
added to the remaining number.
Divisibility by 14
A number divisible by 7 and 2 both will be divisible by 14. To check divisibility by 7 and 2, you will use the same rules as discussed above.
Divisibility by 15
A number divisible by 5 and 3 both will be divisible by 15. Divisibility by 5 and 3 will be the same as discussed above.
Divisibility by 16
A number divisible by 8 and 2 both will be divisible by 16. Divisibility by 8 and 2 will be the same as discussed above.
Divisibility by 17
Osculator for 17 is -5 (negative osculator). Method is the same as discussed in divisibility by 7 and 13.
Divisibility by 18
Number divisible by 9 and 2, will be divisible by 18.
Divisibility by 19
Osculator for 19 is 2 (negative osculator). Method is similar as discussed in divisibility by 7 and 13.
So, Remember that
Osculator of 7 =- 2
Osculator of 13 = + 4
Osculator of 17 =- 5
Osculator of 19 = + 2
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