Divisibility Rules

Divisibility rules are some shortcuts for finding if an integer is divisible by a number without actually doing the whole division process. For example, let's suppose a boy has 531 chocolates, and he has to distribute them among his 9 friends. Dividing 531 by 9, we get no remainder, which means 531 is perfectly divisible by 9.

Table for the Divisibility Rules for 1 to 19

Also Read Divisibility Rules 20 to 30.

Divisibility Rule of 1

As all the numbers are divisible by 1 it doesn't need any test to determine that. Any number k can be written as k×1, thus we can divide k by 1 and still have k left. For example, if 2341 is divided by 1, we have 2341 as the quotient and 0 as the remainder.

Divisibility Rule of 2

A number is divisible by 2 if the last digit of the number is any of the following digits 0, 2, 4, 6, 8. The numbers with the last digits 0, 2, 4, 6, and 8 are called even numbers.

Example: 2580, 4564, 90032 etc. are divisible by 2.

Divisibility Rule of 3

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

Example: 90453 (9 + 0 + 4 +5 + 3 = 21) 21 is divisible by 3. 21 = 3 × 7. Therefore, 90453 is also divisible by 3. 

Divisibility Rule of 4

A number is divisible by 4 if the last two digits are divisible by 4. 

Example: 456832960, here the last two digits are 60 that are divisible by 4 i.e. 15 × 4 = 60. Therefore, the total number is divisible by 4.

Divisibility Rule of 5 

A number is divisible by five if the last digit of that number is either 0 or 5.

Example: 500985, 3456780, 9005643210, 12345678905 etc.

Divisibility Rule of 6

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

Example: 10008, have 8 at one's place so is divisible by 2 and the sum of 1, 0, 0, 0 and 8 gives the total 9 which is divisible by 3. Therefore, 10008 is divisible by 6.

Divisibility Rule of 7

Following are the steps to check the divisibility rule for 7,

1. Take the last digit and then double the last digit.
2. Subtract the result from the remaining number.
3. If the number is 0 or a multiple of 7, then the original number is divisible by 7. Else, it is not divisible by 7.

Example: Consider the number 5497555 to test if it is divisible by 7 or not. Add the last two digits to twice the remaining number and repeat the same process until it reduces to a two-digit number. If the result obtained is divisible by 7 the number is divisible by 7.

• 55 + 2(54975) = 109950 + 55 = 110005
• 05 + 2(1100) = 2200 + 05 = 2205
• 05 + 2(22) = 44 + 5 = 49

Reduced to the two-digit number 49, which is divisible by 7 i.e., 49 = 7 × 7

Divisibility Rule of 8

To check that any number is divisible by 8 we follow the divisibility rule for 8 that states, a number is divisible by 8 if the last three digits of the number are divisible by 8. 

Example: Check 49008 is divisible by 8 or not.

taking last three digits of 49008, '008' which is divisible by 8, therefore, the number 49008 is divisible by 8.

Divisibility Rule of 9

A number is divisible by 9 if the sum of its digits is divisible by 9. In example 90453, when we add the digits, we get the result as 21, which is not divisible by 9, so 90453 is also not divisible by 9.

Example: 909, 5085, 8199, 9369 etc. are divisible by 9. Consider 909 (9 + 0 + 9 = 18). 18 is divisible by 9(18 = 9 × 2). Therefore, 909 is also divisible by 9.

Note A number that is divisible by 9 also divisible by 3, but a number that is divisible by 3 does not have surety that it is divisible by 9.

Example: 18 is divisible by both 3 and 9 but 51 is divisible only by 3, can't be divisible by 9.

Divisibility Rule of 10

A number is divisible by 10 if it has only 0 as its last digit. A number that is divisible by 10 is divisible by 5, but a number that is divisible by 5 may or may not be divisible by 10.10 is divisible by both 5 and 10, but 55 is divisible only by 5, not by 10.

Example: 89540, 3456780, 934260, etc are all divisible by 10.

Divisibility Rule of 11 

To check the divisibility rule for 11, if the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11 completely.

Example: Let us consider a number to test the divisibility with 11, 264482240 mark the even place values and odd place values. Sum up the digits in an even place together and sum up the digits in an odd place together.                        

Now sum up the digits in odd place values i.e., 2 + 4 + 8 + 2 + 0 = 14. To add up the digits in even place values i.e., 6 + 4 + 2 + 4 = 14

Now calculate the difference between the sum of digits in even place values and the sum of digits in odd place values if the difference is divisible by 11 the complete number i.e., 264482240 is divisible by 11. Here the difference is 0, (14-14) which is divisible by 11. Therefore, 264482240 is divisible by 11.

Divisibility Rule of 12

For any number to be divisible by 12, it must be divisible by 3 as well as 4 simultaneously. So, the divisibility rule of 3 and 4 is used together to check whether a number is divisible by 12 or not. 

For example, let's check whether 3276 is divisible by 12 or not.

Divisibility by 3, 3 + 2 + 7 + 6 = 18, which is divisible by 3.

Thus 3276 is divisible by 3.

As 76 is the last two digits of 3276, and 76 is divisible by 4 (76 = 4×19).

Thus, 3276 is divisible by 4 as well.

As 3276 is divisible by 3 and 4 simultaneously, thus 3276 is divisible by 12 as well.

Note: For all the composite numbers such as 14, 16, 18, 20, etc., we can check their divisibility using the divisibility rule of their constituent factors.

Divisibility Rule For 13

To check, if a number is divisible by 13, add 4 times the last digit to the rest of the number and repeat this process until the number becomes two digits. If the result is divisible by 13, then the original number is divisible by 13.

Example: Check whether 333957 is divisible by 13 or not.

Solution:

Unit digit of 333957 is 7,

• (4 × 7) + 33395 = 33423
• (4 × 3) + 3342 = 3354
• (4 × 4) + 335 = 351
• (1 × 4) + 35 = 39
• (1 × 4) + 35 = 39

Reduced to two-digit number 39 is divisible by 13. 

Therefore, 33957 is divisible by 13.

Divisibility Rule of 14

Rule 1: The divisibility rule of 14 states that for any number to be divisible by 14 it should be checked wether that number is divisible by both 2 and 7. If number is divisible by 2 and 7, then automatically that number shall be divisible by 14.

Rule 2: Add the last two digits to twice the number formed by the remaining digits. If the result is divisible by 14, the orginal number is also divisible by 14.

Example: Check if 1064 is divisible by 14 or not.

Solution:

Check for Rule 1:

We need to check if the given number 1064 is divisible by 2and 7 both or not

• 1064 is divisible by 2 as last digit is EVEN number,
• 1064 is also divisbile by 7,
• Hence, 1064 is divisible by 14.

Check for Rule 2:

• Last digit = 64
• Remaining digit = 10
• Twice the remaining digit = 2 x 10 = 20
• Add both numbers = 64 + 20 = 84
• As 84 is divisible by 14 so 1064 is also divisible by 14.

Divisibiilty Rule of 15

A number when divided by 15, is said to be divisible by 15, when it is divisible by both 5 and 3. When both the test of divisibility 5 and 3 are passed we can say that the given number is divisible by 15.

Example: Check 11445 is divisible by 15 or not.

Solution:

We need to checkif the given number is divisible by 3 and 5 Both

Divisibiliy by 3:

• 11445 = 1 + 1 + 4 + 4 + 5 = 15 which is divisible by 3.

Divisibility by 5:

• 11445 ends with unit digit 5 which means that it is divisible by 5.

Since it is divisible by both 3 and 5 so it is also divisible by 15.

Divisibility Rule of 16

The divisibility rule of 16 is unique and has two ways of finding out th results. This is efficinet way to caculate the result of divisibilty rule of 16. The two ways are mentioned below:

Rule 1: If thousand place digit is even:

• Observe the last three digit of the number
• The last dgit are to added to the product of the hundreds place digit multiplied by 4.
• The actual number is divisible by 16 if output is divisible by 16.

Rule 2:If thousand place digit is odd:

• Observe the last three digit of the number
• Add 8 to last three digits.
• The actual number is divisible by 16 if output is divisible by 16.

Example: Check if 21312 is divisible by 16 or not.

Solution:

Given number is 21312.,the digit in thoudsand place is ODD, so the Rule 2 is applied.
The last three digit of the number are 312
So, 312 + 8 = 400
The resultant number 400 is divisible by 16 so the number 21312 is divisible by 16.

Divisibility Rule of 17 

A number is divisible by 17, when dividing it by 17 there is no remainder left. To check, if a number is divisible by 17, subtract 5 times the last digit from the rest of the number and repeat this process until the number becomes two digits.

If the result is divisible by 17, then the original number is also divisible by 17.

 Example: Is 28730 divisible by 17 or not?

Solution:

Unit digit of 28730 is 0,

• 2873 - (5 × 0) = 2873
• 287 - (5 × 3) = 272
• 27 - (5 × 2) = 17

Reduced to two-digit number 17 is divisible by 17. 

Therefore, 28730 is divisible by 17.

Divisibility Rule of 19

To check, if a number is divisible by 19, take its unit digit and multiply it by 2, then add the result to the rest of the number, and repeat this step until the number is reduced to two digits.

If the result is divisible by 19, then the original number is also divisible by 19. Otherwise, the original number is not divisible by 19.

 Example: Is 12635 divisible by 19 or not?

Solution:

Unit digit of 12635 is 5,

• 1263 + (2× 5) = 1273
• 127 + (2 × 3) = 133
• 13 + (2 × 3) = 19

Reduced to two-digit number 19 is divisible by 19. 

Therefore, 12635 is divisible by 19.

Divisibility Tips and Tricks

The following table is the best way to understand the shortcut of the divisibility rules from 2 to 10,

Read More,

• Division
• Easy Division Tricks
• Top 30 Math Tricks for Fast Calculations

Solved Examples of Divisibility Rules

Example 1: Determine the number divisible by 718531.

Solution:

Since, the given number contains 1 in the one's-place, therefore it is clear that it must be divisible either by 3, 7, 9 or 11.

First add all the digits of the given number, 7 + 1 + 8 + 5 + 3 + 1 = 25 which is not divisible by 3 or 9, so 718531 is also not divisible by 3 or 9.

Lets sum up all the even places digits, 3 + 8 + 7 = 18

and now sum up all odd places digits, 1 + 5 + 1 = 7

Now subtract them as:

18 - 7 = 11

Therefore, the given number 718531 is divisible by 11.

Example 2: Use divisibility rules to check whether 572 is divisible by 4 and 8.

Solution:

Divisibility rule for 4 - The last two digits of 572 is 72 (i.e. 4 x 18) is divisible by 4.

Therefore, the given number 572 is divisible by 4.

Divisibility rule for 8 - The last three digits of 572 is,

572 = 2 × 2 × 11 × 13

This implies that, the given number does not contain 8 as its factor, so 572 is not divisible by 8.

Example 3: Check whether the number 21084 is divisible by 8 or not. If not, then find what that number is.

Solution:

The last three digits of the given number 21084 is,

084 or 84 = 2 × 2 × 3 × 7

This implies that, the given number does not contain 8 as its factor, so 21084 is not divisible by 8.

Since, the one's place digit of 21084 is 4 therefore it is clear that 21084 is divisible by 2.

Now, to check the divisibility rule for 4, consider its last two-digits: 84 i.e. 4 × 21.

This implies that, 21084 is divisible by 4.

Hence, 21084 is divisible by 2 and 4.

Example 4: Check if 56355 is divisible by 13, 17 and 19.

Solution:

Example 5: Is 1344 divisible by 2, 3, 4, 5, 6, 7, 8, 9, and 10

Solution: