Divisibility Rules

A divisibility rule is a method that helps check if a number is divisible by another number more quickly, without performing the full division. It saves time by using shortcuts, such as sum rules or patterns, to determine divisibility. For example, let's suppose a boy has 531 chocolates, and he has to distribute them among his 9 friends. Instead of dividing 531 by 9, we can apply the divisibility rule for 9, which says that if the sum of the digits (5 + 3 + 1 = 9) is divisible by 9, then 531 is divisible by 9. Since the sum is 9, 531 is divisible by 9.

Table for the Divisibility Rules for 1 to 19

Also Read Divisibility Rules 20 to 30.

Divisibility Rule of 1

All the numbers are divisible by 1; it doesn't need any test to determine that. Any number, 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 its last digit is one of the following: 0, 2, 4, 6, or 8. Numbers that end in 0, 2, 4, 6, or 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

The following are the steps to check the divisibility rule for 7,

1. Take the last digit of the number and double it.
2. Subtract the result from the remaining number (the number without the last digit).
3. If the result is 0 or a multiple of 7, then the original number is divisible by 7. Otherwise, it is not divisible by 7."

Example: Consider the number 5497555 to test if it is divisible by 7 or not.

To check if a number is divisible by 7, start by taking the last digit, doubling it, and then subtracting the doubled value from the remaining number. Repeat this process with the new number until you get a two-digit result. If the final two-digit number is divisible by 7, the original number is divisible by 7.

• 549755-2 × 5= 549745
• 54974-2 × 5=54964
• 5496-2 × 4= 5488
• 548-2 × 8= 532
• 53-2 ×2=49

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

Divisibility Rule of 8

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 I:s 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 that Ais number that is divisible by 9 is also divisible by 3, but a number that is not divisible by 3 does not have certainty 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, find the alternating sum of the digits of a number. If the result is divisible by 11, then the number is divisible by 11.

Example: Let us consider the number 264482240 to test its divisibility by 11.

Mark the digits in even and odd places. Sum up the digits in the even places and separately sum up the digits in the odd places. Then, subtract the sum of the odd-place digits from the sum of the even-place digits. If the result is divisible by 11 (or is 0), the number is divisible by 11    

Odd positions: 2, 4, 8, 2, 4 (1st, 3rd, 5th, 7th, 9th digits)
sum= 2+4+8+2+4=20

Even positions: 6, 4, 2, 2, 0 (2nd, 4th, 6th, 8th, 10th digits)
sum=6+4+2+2+0=14

Subtract the sum of the even-positioned digits from the sum of the odd-positioned digits:
20-14=6

Since 6 is not divisible by 11, 264482240 is not divisible by 11      

Divisibility Rule of 12

For a number to be divisible by 12, it must be divisible by both 3 and 4 simultaneously. Therefore, the divisibility rules for 3 and 4 are used together to check whether a number is divisible by 12. 

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. Repeat this process until the number is reduced to 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
• (4× 1) + 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 for 14 states that for a number to be divisible by 14, it must be divisible by both 2 and 7.
If a number is divisible by both 2 and 7, then it is automatically 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, then the original number is also divisible by 14.

Example: Check if 1064 is divisible by 14.

Solution:

Check for Rule 1:

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

• 1064 is divisible by 2 as last digit is Even number,
• 1064 is also divisible 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.

Divisibility Rule 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 tests of divisibility by 5 and 3 are passed, we can say that the given number is divisible by 15.

Example: Check if 11445 is divisible by 15

Solution:

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

Divisibility 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 for 16 is unique and can be calculated efficiently using two methods. The two ways to check divisibility by 16 are as follows.

Rule 1: If the last four digits of the number are considered, check whether the number formed by the last three digits (the thousands and hundreds places) is divisible by 16.

• To do this, take the last digit of the number and add it to the product of the hundreds place digit multiplied by 4
• If the result is divisible by 16, then the entire number is divisible by 16.

Rule 2:If the last three digits of the number are considered, add 8 to the last three digits.

• If the sum is divisible by 16, then the entire number is divisible by 16.

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

Solution:

Given number is 21312.,the digit in thousand 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 if, after repeatedly subtracting 5 times the last digit from the remaining part of the number, the result is divisible by 17.

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 the unit digit (the last digit) of the number and multiply it by 2. Then, add the result to the remaining part of the number (the number without the last digit). Repeat this process with the new number until you are left with a two-digit number. If the final two-digit number is divisible by 19, then the original number is divisible by 19. However, if the final two-digit number is not divisible by 19, then 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 a shortcut way to understand the shortcut for divisibility. Divisibility shortcut 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 numbers 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 rule for 8 states that the last three digits of the number should be divisible by 8.

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 are 084, and 084 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, Rules Divisibility and 19.

Solution:

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

Solution:

Practice Problems on Divisibility Rules :-

Question 1.- Is the number 3328 divisible by 8 or not?
Question 2.- Is the number 1001 divisible by 11 or not?
Question 3.- Is the number 1190 divisible by 17,numbers and is 1007 divisible by 19 or not?
Question 4.- Is the number 56173 divisible by 13 and is 60494 divisible by 14 or not?

Answer:-
1. Yes.
2. Yes.
3. Yes, Yes
4. Yes, Yes