Divisibility Rules

In order to make the division process simple and quicker, divisibility rules or divisibility tests are used. If one has the knowledge of the divisibility rules worksheet, they can easily do the simple calculations. The known divisibility tests are from 2 to 20. The divisibility rule for a certain number can tell what is completely divisible by that number. What are the divisibility rules? Let’s understand it in more detail.

What are Divisibility Rules?

The divisibility rules are introduced as a shortcut to find if an integer is divisible by a number without actually doing the whole division process. The divisibility rules can help determine the prime factorization relatively fast. Let’s understand divisibility rules with examples. Suppose a boy has 537 chocolates, and he has to distribute them among his 9 friends. How can he do it? Dividing 537 by 9, he has left with some chocolates (remainder), which means 537 is not divisible by 9 exactly. Dividing is simple to check that the number is exactly divided by the divisor i.e., Remainder is 0 or not when one has 2 or 3-digit numbers. If the number is too large, it takes a long time to perform the actual division. How can we know whether a number is divisible by a particular divisor or not? Here comes the concept of divisibility rules: a quick, easy, and most straightforward way to find out the divisibility of a number by a particular divisor. Let’s check all important divisibility rules,

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 Rules 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 Rules 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.

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 Rules 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 Rules 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 Rules 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 Rules 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 Rules of 8

To check the divisibility rule for 11, if a number is divisible by 8 its last three digits should be divisible by 8. 

Example: 008 which is divisible by 8, therefore, the total number is divisible by 8.

Divisibility Rule of 7

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

1. Removing the last digit and then double the number.
2. Subtract it from the remaining number.
3. If the number is 0 or a multiple of 7, then the original number is divisible by 7. Else, according to the divisibility test, 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 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: Consider a number to test the divisibility with 4 and 8, 456832960 mark the even place values and odd place values. Sum up the digits in even place values together and sum up the digits in odd place values together.                        

Digits	Place Value
4	0
5	1
6	2
8	3
3	4
2	5
9	6
6	7
0	8

Now sum up the digits in even place values ie 0 + 2 + 4 + 6 + 8 = 4 + 6 + 3 + 9 + 0 = 22. To add up the digits in odd place values i.e.

1+ 3 + 5 + 7 = 5 + 8 + 2 + 6 = 21

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., 456832960 is divisible by 11. Here the difference is 1, (22-21) divisible by 11. Therefore, 456832960 is divisible by 11.

Divisibility Rules Tips and Tricks

Below given table is the best way to understand the shortcut of the divisibility rules from 2 to 10,

Some More Divisibility Rules

Co-primes are the pair of numbers that have 1 as the common factor. If the number is divisible by such co-primes, the number is also a divisible by-product of the co-primes. For instance, 80 is divisible by both 4 and 5 they are co-primes that have only 1 as the common factor, so the number is also divisible by 20, the product of 4 and 5

1. 21 = 3 × 7                          
2. 12 = 3 × 4
3. 22 =11 × 2                        
4. 14 = 2 × 7
5. 15 = 3 × 5                            
6. 30 = 3 × 10
7. 18 = 2 × 9
8. 28 = 4 × 7
9. 26 =13 × 2

If a number is divisible by some numbers, say X, that number is also divisible by factors of x. 

Example: If a number is divisible by 40 then it is divisible by its factors i.e: 5, 10, 2, 4, 8, 20.

Divisibility Rule For 13

To check, if a number is divisible by 13 add 4 times the last digit of the number 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: 333957

(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.

Summary of Divisibility Rules

• Divisibility rule is a shortcut to analyze whether an integer is completely divisible by a number without actually doing the factorization.
• Divisibility rules worksheet

Divisibility by number	Divisibility rule
Divisibility by 2	The last digit should be even.
Divisibility by 3	The sum of the digits should be divisible by 3.
Divisibility by 4	The last two digits should be divisible by 4.
Divisibility by 5	The last digit should either be 0 or 5.
Divisibility by 6	The number should be divisible by both 2 and 3.
Divisibility by 7	The double of the last digit, when subtracted by the rest of the number, the difference obtained should be divisible by 7.
Divisibility by 8	The last three digits should be divisible by 8.
Divisibility by 9	The sum of the digits should be divisible by 9.
Divisibility by 10	The last digit should be 0.
Divisibility by 11	The difference of the alternating sum of digits should be divisible by 12.
Divisibility by 12	The number should be divisible by both 3 and 4.

Solved Examples on 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: Test 224 for divisibility by 7.

Solution:

First double the last number i.e 4 of the given number (224) ⇒ 2 × 4 = 8.

Subtract this number from the rest of the digits ⇒ 22 – 8 = 14.

This implies that, the obtained number is divisible rule for 7, hence the given number 224 is divisible by 7.

Example 5: Check on 2795 for divisibility by 13.

Solution:

The last number of the given number i.e. 2795 is 5, 

Multiply 4 by 5 and add to the rest of the digits as:

⇒ 279 + (5 × 4) 

= 299.

Similarly, again multiply 4 by the last digit (i.e. 9) of the obtained three-digit number (i.e. 299) and add to the rest of the digits as:

⇒ 29 + (9 × 4) 

= 65.

Now, a two-digit number is obtained i.e. 65 = 5 × 13.

Hence, 2795 is divisible by 13.

FAQs on Divisibility Rules

Question 1: What are divisibility rules?

Answer:

Divisibility rules are the trick to finding out whether a number is divisible by another number without doing the actual calculation.

Question 2: What are the divisibility rules for 3 and 9?

Answer:

A number is divisible by 3 if the sum of its digits is divisible by 3. Similarly, A number is divisible by 9 if the sum of its digits is divisible by 9. For example, 39 is divisible by 3, as 3 + 9 = 12, which is divisible by 3. Therefore, 39 is divisible by 3. 2397 is divisible by 9, as 2 + 3 + 9 + 7 = 21, which is divisible by 9.

Question 3: What is the divisibility rule for 13?

Answer:

The divisibility rule for 13 is to add four times the last digit of the number to the remaining number until a two-digit number is obtained. If the two-digit number is divisible by 13, the number too is divisible by 13.

For example, Take 1001

Adding 4 to 100, 100 + 4 = 104

Adding 16 to 10, 10 + 16 = 26

Now since 26 is divisible by 13, 1000 is divisible by 13.

Related Resources

Class 8 Maths Notes for Divisibility Rules

Class 8 Maths NCERT Solutions for Divisibility Rule

RD Sharma Class 8 Solutions for Divisibility Rules