Divisibility Rules (Divisibility Tests): Checking the divisibility for the complex numbers can be difficult at times. Do you agree or not? If yes, then this guide is completely on the divisibility rules to make it simple for all students and candidates who are preparing for competitive exams and board exams.
A page with all 1 to 25 divisibility rules will guide you through your calculations and make you learn the logic and tricks to solve further. Continue your read and get the direct links to all divisibility tests with examples and other benefits like definitions, formulas, tips and tricks, etc.
Here are the free links that support you while calculating the divisibility of 1 to 25. This main divisibility criteria list helps you know the given number is divisible by another number without performing any long division calculations. Let’s click on the required divisibility rule link and learn the trick of calculating complex divisibilities.
In maths, the divisibility test or division rules is the shorthand technique or trick used to check whether a large number is divisible by another number without using the division method. When you divide a number with another number and a quotient is a whole number and the remainder is zero, then that number is said to be divisible by another number.
There is no specific trick or condition for the divisibility rule of 1 because every number is divisible by 1 and it gives the number itself. For instance, 5638 is divisible by 1 and 8 is divisible by 1.
The test of divisibility by 2 is any number should be even or its last digit needs to be an even number i.e., 2,4,6,8 including 0, then the number is divisible by 2. For example, 488 is divisible by 2, as the last digit 8 is divisible by 2 and it is an even number.
When the sum of the digits of a given number is divisible by 3 then the number is completely divisible by 3 as per the divisibility test of 3. Hence, the digital root of the number is 0 or 3 or 6 or 9. For example, 576 is divisible by 3 or not. To check it, let's do the summation of all digits ie., 5+7+6 = 18. 18 is divisible by 3 so 576 is divisible by 3.
If the last two digits are zero or divisible by 4 then the number is divisible by 4. Now, let's see if 784 is divisible by 4 or not. The last two digits are 84 and it is divisible by 4. So 784 is divisible by 4.
The number can be divisible by 5 if it ends with 0 or 5. For example, let's check if 457 is divisible by 5 or not. The last digit of the given number is 7 so it is not divisible by 5. Also, see if this large number is divisible by 5 or not ie., 396524850. This is divisible by 5 because 0 is the last digit of this number.
The divisibility rule of 6 is stated as the number is divisible by 6 when it is divisible by both 2 and 3. Check whether the number is even or not and whether the sum of digits is divisible by 3 or not. If two conditions are satisfied then it is divisible by 6.
For Example, 360 is the number that can be divisible by 6 or not. 360 is the even number and it ends with 0 so the number is divisible by 2. The sum of digits is 9, it is also divisible by 3. Therefore, 360 is divisible by 6.
When you want to check the given number is divisible by 7 apply the divisibility rule of 7 which states to double the unit digit and then subtract the result with the remaining digits. Repeat the process until you are left with two digits or a single digit. If the number is 0 or a multiple of 7 then it is divisible by 7.
For instance, find 987 is divisible by 7 yes or no? After applying the divisibility rule for 7, double the unit digit 7 and subtract with 98, ie., 98 - 14 = 84. The result is a multiple of 7. Hence 987 is divisible by 7.
When the last three digits are divisible by 8 or the last three digits are zeros, then the given number is divisible by 8 completely.
For instance, apply the divisibility rule of 8 on the number 46598. 598 are the last three digits. Now, check whether it is divisible by 8 or not. Here, the hundred digit is odd so we have to apply the odd rule ie., the last two digits plus 4 is divisible by 8, then 46598 is divisible by 8. Here, 98 + 4 = 102 and it is not divisible by 8, so 46598 is not divisible by 8.
The test for divisibility by 9 is the same as the divisibility rule for 3. If the sum of all digits is divisible by 9 then the given number is divisible by 9.
For example, take the number ie., 86240. Do the summation of all digits, 8+6+2+4+0 = 20. The number 20 is not divisible by 9, therefore 86240 is not divisible by 9.
Divisibility test for 10 is the number ends with 0 is divisible by 10. For example, 30, 1000, 7000, etc. are divisible by 10.
The number is divisible by 11 when the difference of the sum of alternative digits of a given number is divisible by 11.
For instance, 4568 is divisible by 11 as 4+6 = 10 and 5+8 = 13 now 10-13 = -3, which is not divisible by 11.
Divisibility Rule of 12 with Example
Divisibility rule by 12 states that if the number is divisible by both 3 and 4 then the number is divisible by 12. For example, 674 is divisible by 12 or not. Now, add all the digits ie., 6+7+4 = 17. 17 is not divisible by 3. Let's check that 674 is divisible by 4, and the last two digits 74 are not divisible by 4. So, 674 is not divisible by 12.
When you multiply the last digit by 4 and add it to the rest of the number, after adding the result should be divisible by 13, then the number is divisible by 13.
For example, 409 is divisible by 13, multiply 9 by 4 and add it to 40 ie., 40 + (9x4) = 76, 76 is not divisible by 13. Therefore 409 is not divisible by 13.
Divisibility Rule of 14 says when the given number is divisible by 2 and 7 both then the number is divisible by 14. 2 and 7 are prime factors of 14.
For example, 7632 is divisible by 14 or not. The last digit 2 is an even number so it is divisible by 2. Now, check if it is divisible by 7 or not, to do that:
When the number is exactly divisible by 15, then it should divisible by both 3 and 5.
For example, 540 is divisible by 15.
Hence, 540 is not divisible by 15.
If the last four digits of the number formed are divisible by 16 when a thousand place digit is an even number, then the given number is divisible by 16. If a thousand place digit is odd then the number formed by the last three digits plus 8 is divisible by 16.
For instance, 162448 is the number to check is divisible by 16 or not. Now, see a thousand digits ie., 2 is even or not, it is an even number. Let's check the four last formed digits ie, 2448 formed are divisible by 16. Hence, the number 162448 is divisible by 16.
Divisibility test by 17 is like multiplying the last digit by 5 and minus it from the rest of the number, and the end result should be divisible by 17.
For instance, check if 345 is divisible by 17 or not. First, multiply 5 by 5 times ie., 5x5 = 25, and now subtract 25 from the remaining number in a given number ie., 34 - 25 = 9. Hence, 9 is not divisible by 17.
Divisibility Rule of 18 with Example
As per the divisibility rule of 18, if any number is divisible by 2 and 9 both then the given number is exactly divisible by 18. When the number satisfies the two conditions then it is said to be divisible by 18.
For example, 6589 is divisible by 18 or not. The last digit of the number is 9 which is not an even number so it is not divisible by 2. Now, check it is divisible by 9 or not by summation of all digits ie., 6+5+8+9 = 28. 28 is not divisible by 9. Therefore, the given number 6589 is not divisible by 18.
The divisibility rule of 19 is having two rules to be followed to check whether the number is divisible by 19 or not.
Rule 1: If the last digit is multiplied by 2 and added to the remaining number in a given number and the result is divisible by 19 then the number is exactly divisible by 19 when it is a small number.
Rule 2: If the given number is big, then multiply the last digit by 4 and add it remaining number, it should be divisible by 19.
For example, let's take 456 is divisible by 19. Initially, multiply the 6 by 2 ie., 6x2 = 12. Now, add 45 and 12 ie., 45+12 = 57. 57 is divisible by 19 so 456 is divisible by 19.
If the formed number with the last two digits is divisible by 20 then the given number is divisible by 20. One more version to test the divisibility of 20 is if the ten’s digit is even and the unit place is 0 then it is divisible by 20.
For example, 45890 is divisible by 20 or not. Take the last two digits into consideration and check whether the ten’s place digit is even or not. 9 is not an even number although the unit digit is 0, the given number 45890 is not divisible by 20.
To test the divisibility of 21 for any number, consider the unit digit and multiply it by 2 and subtract it from the remaining number. The end result must be divisible by 21 so that the given number is divisible by 21.
For example, 4563 is divisible by 21 or not. Take the last digit (3) and multiply by 2 ie., 3x2 = 6. Now subtract 456 - 6 = 450.
Repeat the first two steps for the number, 450 ie. 45 - (0x2) = 45-0 = 45 and 45 is not divisible by 21.
Hence, 4563 is not divisible by 21.
According to the divisibility rule for 22, the number is exactly divisible by 22 if it is an even number and the result of alternate subtracting and adding the digits in the number is divisible by 11.
For instance, let's see if 6034 is divisible by 22 or not. The number is an even number and 6-0+3-4 = 6-1 = 5. Since 5 is not divisible by 11, then 6034 is not divisible by 22.
If the last digit is multiplied by 7 and added to the remaining number in a given number then the result should be divisible by 23 to say the given number is divisible by 23 as per the divisibility rule for 23.
For instance, 657 is divisible by 23 or not. To check it, apply the divisibility rule of 23 ie, 65 + 7x7 = 65 + 49 = 114. It is not divisible by 23 hence 657 is not divisible by 23.
The divisibility rule of 24 states if the number is divisible by both 3 and 8 then it is divisible by 24. Remember both conditions should be satisfied to get divisible by 24.
For example, 45123 is divisible by 24 or not.
Check by 3: Do add all the digits ie., 4+5+1+2+3 = 15. 15 is divisible by 3.
Check by 8: Here the last three digits e., 123 are not divisible by 8.
Hence, 45123 is not divisible by 24.
To test if the number is divisible by 25, you need to check whether the last two digits are formed as 00, 25, 50, or 75.
For example, 7655 is divisible by 25 or not. The formed last two digits are 55 so it is divisible by 25. Therefore, 7655 is divisible by 25.
Divisibility by number |
Divisibility Rule |
Divisible by 2 |
A number that is even or a number whose last digit is an even number i.e. 0, 2, 4, 6, and 8. |
Divisible by 3 |
The sum of all the digits of the number should be divisible by 3. |
Divisible by 4 |
The number formed by the last two digits of the number should be divisible by 4 or the last two digits be zeros ‘00’. |
Divisible by 5 |
Numbers having 0 or 5 at the units place digit. |
Divisible by 6 |
A number that is divisible by both 2 and 3. |
Divisible by 7 |
Subtracting twice the last digit of the number from the remaining digits gives a multiple of 7. |
Divisible by 8 |
The number formed by the last three digits of the number should be divisible by 8 or should be 000. |
Divisible by 9 |
The sum of all the digits of the number should be divisible by 9. |
Divisible by 10 |
Any number whose one place digit is 0. |
Divisible by 11 |
The difference between the sums of the alternative digits of a number is divisible by 11. |
Divisible by 12 |
A number that is divisible by both 3 and 4. |
1. How many divisibility rules are there in math?
In math, there are so many divisibility rules but basically, we have rules of divisibility from 1 to 20. Later, we discussed divisibility rules for 21 to 25 on our page by identifying the pattern of number multiples from other divisibility tests.
2. What is the formula of divisibility?
If we found the division problem in an equation using our division algorithm, and r=0, then the following equation is used ie., a = bq.
3. What is divisibility with example?
In mathematics, divisibility means a number that is evenly divided by another number, with no remainder and the whole number as a quotient. For example, 30 is divisible by 3 as its quotient is 10 and the remainder is 0.
We hope the information on divisibility rules from 1 to 25 helped you greatly with your complex division calculations. If you need to get more details about each divisibility test then go for the above links and make use of them too. Remember each trick to memorize the divisibility rules to solve the divisible of large numbers.