Divisibility Rules from 1 to 13, 17 & 23 with Easy Examples

Division is a basic math operation where we split one number by another. When we say a number “p” is divisible by another number “q”, it means dividing p by q leaves no remainder. In this case, p is called the dividend (the number being divided), and q is the divisor (the number we divide by). To make division quicker and easier, we use divisibility rules. These are simple checks or tricks that help us know if one number can be divided by another without actually doing the full division. For example, if a number ends in 0, it is divisible by 10. These rules save time and are very helpful in solving math problems, especially when working with large numbers. In this topic, you will learn the divisibility rules for numbers from 1 to 12 and for some prime numbers, along with useful examples to help you understand better.

Divisibility Rules

Divisibility rules are easy tricks in mathematics that help us check if one number can be divided by another without actually doing long division. These rules are made to help us work with large numbers faster and save time, especially when solving math problems. For example, if we know the divisibility rules from 1 to 20, we can quickly figure out if a number has certain factors or if it can be divided evenly by another number.

This is very helpful during exams or while solving quantitative problems, as it reduces the need for long calculations. For instance, if a number ends in 0, we can instantly say it’s divisible by 10. Learning and remembering these rules allows us to quickly check divisibility in our head and find factors of large numbers. A divisibility rules chart makes this process even easier by giving all the rules in one place for quick reference.

Divisibility Rules From 1 to 13

Having the basic idea of what are the divisibility rules, let us understand the rules for 1 to 12 with a detailed explanation.

In division operation, Dividend = (Divisor x Quotient) + Remainder.

If one number is completely divisible by another number then in such a case the quotient would be a whole number and the remainder will be zero.

When a non-zero integer, say; P, divides another integer, say; Q and provides a remainder as zero, then we say that P is a factor of Q.

It is not always possible that every number is completely divisible by every other number and the remainder is equal to zero. These divisibility rules for the specific numbers would help us to decide the exact divisors of a number simply by evaluating the digits of the number.

Divisibility Rule of 1

Rule: Every number is divisible by one irrespective of the number, its type or how big that number is. Hence, there is no such particular divisibility rule for 1.

Example: 40 is divisible by 1, 400 is divisible by 1 and 40000 is also divisible by 1.

Divisibility Rule of 2

Rule: A number is divisible by 2 if it is an even number i.e., if the number ends(units place) with 2, 4, 6, 8 or 0.

Example: 42, 126, 284, 3642, 50222, 876898000 and so on are all divisible by 2.

Note: For huge numbers like876432984, to check for the divisibility no need to completely divide the number. Just check the last digit. Here the last digit is 4 and is divisible by 2, thus the complete number is divisible by 2.

Divisibility Rules for 3

Rule: Any number is divisible by three (3) if the aggregate of the digits of the number is a multiple of 3 or the aggregate of the digits of the number is divisible by 3.

Example: Consider the number348. To check for the divisibility with 3, add the digits of the number. 3+4+8=15. As 15 is divisible by 3 therefore the complete number is also divisible by 3.

Divisibility Rule of 4

Rule: Any number with three or more digits is divisible by 4 if the number formed by taking the digit in the ones and tens place i.e. the last two digits of the number are divisible by four(4).

Example: For the given number;3124. To check for the divisibility by 4, consider the last two digits and check if it’s divisible by 4. Here it is 24 and 24 is divisible by 4(24/4=6). Thus, the complete number is 3124 and is also divisible by 4.

Check out this article on Binary Numbers here.

Divisibility Rule of 5

Rule: A number with n number of digits is divisible by 5 if the last digit of the number is 0 or 5.

Examples: 1000, 1004205, 5845, etc are all divisible by 5.

Divisibility Rule of 6

Rule: Any number on the number line is divisible by 6 if it passes the divisibility test of both 2 and 3. That is, a number is divisible by 6 if it is divisible by both 2 and 3.

Example: Consider the number;816, to check if it is divisible by 6 we check if it is divisible by 2 and 3.

The number 816 is divisible by 2, as it fulfills the divisibility condition of 2.

816/2=408

The number is also divisible by 3.

As, 8+1+6=15 and 15/3=5

As 816 is divisible by 2 and 3, the number is further divisible by 6.

Divisibility Rules for 7

Rule: A given number is divisible by 7 if the difference between the number formed after excluding its last digit(unit place) and the double of the last digit is a number divisible by 7. If in case the resultant is not divisible by 7, then the complete number is not divisible by 7.

Example: Consider the number; 308. check if it is divisible by 7.

Following the rule:

Double of the last digit=16

Subtracting the result from the rest of the number; 30-16=14

14 is a multiple of 7, hence the number is divisible by 7.

Divisibility Rule of 8

Rule: A number that has 4 or more digits is divisible by 8 if the number formed by the last 3 digits of the number is divisible by 8.

Example: Consider the number17320. Here the number formed by the last three digits(hundreds, tens, ones)=320.

320/8=40

As the number formed by the last three is divisible by 8, the number17320 is also divisible by 8.

Also learn the various types of Number Series here.

Divisibility Rule of 9

Rule: The divisibility for 9 is the same as the divisibility rule for 3. The rule states that, if the aggregate of all the digits of the given number is divisible by 9, then the entire number is divisible by 9.

Example: Consider the number;5769. check if it is divisible by 9.

As per the rule, the addition of all the elements=5+7+6+9=27

27 is divisible by 9, therefore the number 5769 is also divisible by 9.

Divisibility Rule of 10

Rule: Any number from the number system that has zero (0) in its unit’s place is divisible by 10.

Examples: 10, 790, 30, 26970, 196570, 1070, etc are all divisible by 10.

Divisibility Rules for 11

Rule: The divisibility rule for 11 is somewhat different from the other rules. It says that; if the difference between the sum of the digits at odd positions starting from the right and the sum of the digits at the even positions starting from the right of the number is either 0 or divisible by 11, then the number itself is divisible by zero.

Example: Consider the number1826. Check if it is divisible by 11.

As per the rule;

Starting from the right of the expression pic the odd and even place digits.

Odd place digit sum=8+6=14

Even place digit sum=2+1=3

Difference between both the results=14-3=11

As the result of the difference is divisible by 11 the number1826 is also divisible by 11.

Divisibility Rule of 12

Rule: For any given number to be divisible by 12 the number must be divisible by both 3 and 4. That is, a number must satisfy the divisibility rule of both 3 and 4 to be divisible by 12.

Example: For the given number; 864.

864 is divisible by 4 as the last two-digit(64) is divisible by 4.

The addition of the digits in the number=8+6+4=18. The number 18 is divisible by 3 hence, 864 is also divisible by 3.

The number 864 is divisible by both 3 and 4. Therefore it is divisible by 12 as well.

Divisibility Rule of 13

Rule: The divisibility rule of 13 states that; Add the unit place digit after multiplication with 4 to the remaining number to the left of the digit at units place. If the result of the addition is divisible by 13, then the complete number is also divisible by 13.

Example: For the given number338, check if it is divisible by 13.

As per the rule multiplying the unit digit by 4, we get; 8×4=32

Now add the rest of the number with 32; 33+32=65

The resultant 65 is divisible by 13, thus the entire number is also divisible by 13.

Divisibility Rule for Prime Numbers

Previously, we learnt the divisibility rule of 1 to 13, where we cover the divisibility of prime numbers like 2, 3, 5, 7, 11 and 13. Let’s learn the divisibility rule for other prime numbers like 17, 19 and 23.

Divisibility Rule of 17

Rule: To check for the divisibility of a number with 17, first multiply the unit place digit with 5. Next, subtract the resultant outcome from the remaining digits. If the resultant is divisible by 17, then the complete number is also divisible by 17.

Example: For the provided number, 442. Check if it is divisible by 17.

As per the rule, multiply the unit place digit with 5; 2 × 5=10

Subtract the resultant outcome from the remaining digits; 44 − 10 = 34

As 34 is divisible by 17 the number 442 is also divisible by 17.

Divisibility Rule of 19

Rule: For a number to be divisible by 19; first multiply the digit at the one’s place by 2. Then, add the number formed by excluding the unit place digit with the previous product obtained. If the result is divisible by 19 then the number will also be divisible by 19.

Example: Check the divisibility of the number for 19; 703

As per the rule;

Multiply the digit at the unit place with 2; 3 x 2=6

Add the rest of the number excluding the unit place digit with the product obtained; 70 + (3 x 2) = 76

76 is divisible by 19, thus the number 703 is also divisible by 19.

Divisibility Rule of 23

Rule: For a number to be divisible by 23; Multiply the last digit by 7 and add the result to the number formed by removing the last digit. If the final result is divisible by 23, then the given number will also be divisible by 23.

Example: For the number 414, check the divisibility with 23.

As per the rules multiply the unit digit by 7; 4 x 7=28

Add the result with the remaining value; 41+28=69

69 is divisible by 23, therefore the number 414 is also divisible by 23.

Learn about the various Natural Numbers here.

Divisibility Tips and Tricks 

Use these simple tricks to quickly check if a number can be divided by another without doing full division:

Divisible By	Easy Rule	Example
2	If the last digit is 0, 2, 4, 6, or 8, the number is divisible by 2.	132, 100, 12
3	Add up all the digits. If the total can be divided by 3, then the number can too.	4764 → 4+7+6+4 = 21 → 21 ÷ 3 = 7
4	Look at the last two digits. If they form a number you can divide by 4, it works.	1924 → 24 ÷ 4 = 6
5	If the last digit is a 0 or a 5, it’s divisible by 5.	255, 1970
6	The number must follow the rules for both 2 and 3.	7782 → It’s even, and 7+7+8+2 = 24 → 24 ÷ 3 = 8
7	Double the last digit and subtract it from the rest of the number. If the result can be divided by 7, it works.	385 → 38 − (2×5) = 28 → 28 ÷ 7 = 4
8	Look at the last three digits. If that number can be divided by 8, it works.	1320 → 320 ÷ 8 = 40
9	Add all the digits. If the total can be divided by 9, so can the number.	288 → 2+8+8 = 18 → 18 ÷ 9 = 2
10	If the number ends in 0, it’s divisible by 10.	8070

Divisibility Rules Chart

In the previous headers we learnt the divisibility rule for different numbers from 1-12 along with prime numbers like 13, 17, 19 and 23. It is sometimes difficult to remember the rules step by step. Thus below is the summary chart for all the rules discussed so far for a quick revision.

Divisibility Rule by Number	Divisibility Rule
Divisibility Rule of 1	There is no such specific rule as any given number is divisible by 1.
Divisibility Rule of 2	All numbers whose last digit is even are divisible by 2.
Divisibility Rule of 3	If the sum of all the digits is divisible by the 3, then the number is also divisible by 3.
Divisibility Rule of 4	When the last two digits from the number are divisible by 4, then the entire number is also divisible.
Divisibility Rule of 5	All the numbers with the last digit as 0 or 5 are divisible by 5.
Divisibility Rule of 6	Any number that follows the divisibility test of both 2 and 3 is divisible by 6.
Divisibility Rule of 7	The difference of twice the last digit from the number from the remaining digits should be divisible by 7.
Divisibility Rule of 8	The last three digits of the given number should be divisible by 8, for the complete number to be divisible by 8.
Divisibility Rule of 9	The total of all the digits of the given must be divisible by 9, for the number to be divisible by 9.
Divisibility Rule of 10	All types of numbers whose one place digit(last digit from the right) is 0 are divisible by 10.
Divisibility Rule of 11	The difference of the aggregate of the even digits and odd digits of the given number should be divisible by 11, for the complete number to be divisible by 11
Divisibility Rule of 12	The number that fulfills the divisibility rule of both 3 and 4 is divisible by 12 as well.
Divisibility Rule of 13	Sum of the product of the last digit with 4 and the number formed by remaining digits, should be divisible by 13.
Divisibility Rule of 17	Multiplication of the last digit by 5 and subtracting the resultant product from the remaining digits should result in a number that is a factor of 17.
Divisibility Rule of 19	The sum of the double of the last digit and the number formed by remaining digits should be a multiple of 19.
Divisibility Rule of 23	The sum of the product of the last digit with 7 and the rest of the number, must be a factor of 23.

Solved Examples of Divisibility Rules

Knowing all about the math divisibility rules along with the division rules chart, let us rehearse some solved examples for more practice on the topic.

Example 1: Check if the below numbers are divisible by 11 or not?

1. A) 3839
2. B) 4518

Solution: The divisibility rule for 11 says that; the difference of the aggregate of the alternate digits of the given number should be divisible by 11, for the complete number to be divisible by 11.

For the number 3839

Number of digits = 4

The sum of odd numbers from the RHS=9+8=17

The sum of even numbers from the RHS=3+3=6

Difference between both the result=17-6=11

The resultant is divisible by 11, thus the number 3839 will also be divisible by 11.

Now consider the number; 4518

Using the same approach as above.

Number of digits = 4

The sum of odd numbers from the RHS=8+5=13

The sum of even numbers from the RHS=1+4=5

Difference between both the result=13-5=8

As, the resultant is not divisible by 11, thus the number 4518 is also not divisible by 11.

Also, learn about Even Numbers.

Solved Example 2: Form the given set of numbers, pick the ones that are divisible by 4.

846 124 636 394

Solution: The divisibility rule of 4 states that; when the number formed by the last two digits from the number are divisible by 4, then the entire number is also divisible.

From the given options, only the numbers formed by the last two digits of 124 and 636 are divisible by 4. Hence only these numbers are divisible by 4.

Learn more about Odd Numbers here.

Solved Example 3: If the aggregate of all the digits of a given number is divisible by 9, then which of the below options is true?

1. a) The number is divisible by 4 also.
2. b) The number is divisible by 3 and 9 both.
3. c) The number is divisible by 6.
4. d) None of these

Solution: If the aggregate of all the digits of a given number is divisible by 9, this implies that the number itself is divisible by 9. This further implies that that number will be divisible by 3 also, as 3 is a factor of 9.

Hence option b is correct.

Rest for the option a and c, it is not possible to state that if a number is divisible by 9, then it will be divisible by 4 or 6. As the divisibility criteria of 4 and 6 are different.

Check out this article on Addition and Subtraction of Algebraic Expressions.

Example 4: On certain months of a year the number of test series brought by the students preparing for bank exams is as follows:

June:494 test series were sold

August: 259 test series were sold

Which of the below statements is true?

(A) The number of test series sold in June is divisible by 13.

(B) The number of test series sold in the month of August is divisible by 7.

Solution: Start with Statement A;

The number of test series sold in June is divisible by 13.

In June 494 test series were sold.

Applying the divisibility rule of 13, we get:

49+(4×4)=65

The outcome is divisible by 13 and so will be the number 494. Therefore the statement is true.

Statement B;

The number of test series sold in the month of august is divisible by 7.

In August 259 test series were sold.

Applying the divisibility rule of 7, we get:

25-(9×2)=7

The result is divisible by 7 and so will be the number 259. Therefore the statement is true.

If you are checking Divisibility Rules article, also check the related maths articles in the table below:
Difference between exponent and power	Multiplicative inverse
Markup	Consecutive integers
Factors of 54	Factors of 32

 

Important Links
NEET Exam
NEET Previous Year Question Papers	NEET Mock Test	NEET Syllabus
CUET Exam
CUET Previous Year Question Papers	CUET Mock Test	CUET Syllabus
JEE Main Exam
JEE Main Previous Year Question Papers	JEE Main Mock Test	JEE Main Syllabus
JEE Advanced Exam
JEE Advanced Previous Year Question Papers	JEE Advanced Mock Test	JEE Advanced Syllabus