This document discusses 20 divisibility rules in mathematics. It explains that divisibility rules provide shortcuts to determine if a number is divisible by another number without long division. Then it proceeds to define each divisibility rule from 1 to 20, giving examples for each. The rules cover divisibility by numbers up to 20, with some rules like 7, 11, and 13 being more complex than others.

1. 1 – 20 Divisibility Rules in Mathematics
One interesting fact about Mathematics is that it comes with tips and
tricks. Learning its rules and tricks itself is a skill. It’s the responsibility of
every math expert, whether a school teacher or private math tutor, to
teach rules of mathematics to every student. In this article, we are going
to discover divisibility rules of mathematics. Such rules, shorthand
techniques, and tricks make complex math concepts a bit handy for some
of us.
To quickly determine if a number is divisible by 1 to 20 natural numbers
without doing long division or lengthy calculations, students must know
everything about divisibility rules. Learning these rules is helpful for
quick mental arithmetic and saves time during competitive exams when
numbers are large. Divisibility rules are also known as divisibility tests.
They make the division procedure easy and quick. Let us learn and
understand all divisibility rules in-depth with solved examples.

2. What are Divisibility Rules?
In mathematics, divisibility rules are a set of methods or rules to check
whether the given number or integer is completely divisible by the other
number without performing lengthy calculations or the wh ole division
process.
In 1962, a popular science and mathematics writer named Martin
Gardner explained and discussed the divisibility rules. He stated that such
rules are made to reduce large number fractions down to the lowest
terms. There are actual numbers completely divisible by other numbers
without leaving a remainder. However, some might divide the number but
leave a remainder other than zero. Therefore, divisibility rules were made
to find out the actual divisor or a number by considering some facts.
Definition of Divisibility Rules
By definition, a whole number m divides another whole number or integer
n if and only if you can find a nonzero integer y such that m x y = n where
the remainder will be zero.
For instance, 8 is divisible by 2 because 2 x 4 = 8
By considering the digits of a number, you can quickly determine its
actual divisor. There are only 1 to 13 divisibility rules in mathematics,
which we have explained in this a rticle with examples. Rules of numbers
less than 5 are quite easy to understand; however, the divisibility rules of
7, 11, and 13 are complex and hard to understand. Let’s go through all the
rules with examples and understand how to divide the numbers quic kly
in-depth.
Rule No. 1: Divisibility by 1

3. Not all rules of mathematics have a condition. That’s the same case with
the divisibility rule for 1. All numbers, whether small or large, are
divisible by 1. Numbers divided by 1 give the number itself with zero
remainders. In a nutshell, all numbers or i ntegers are divisible by 1.
For Example
Take any number like 5 or 6000, and they are completely divisible by 1.
Rule No. 2: Divisibility by 2
Any number has 0, 2, 4, 6, and 8 as their last digits are divisible by 2. The
basic divisibility rule of 2 is that all even numbers are completely divisible
by 2.
For Example
Let’s take any even number and check if its last digit is divisible by 2 or
not.
For instance, consider the number 668. Its last digit is 8, which is
completely divisible by 2. Thus, 668 is also divisible by 2.
Rule No. 3: Divisibility by 3
The rule of divisibility by 3 states that if the sum of the digits of the given
number (dividend) is completely divisible by 3, then the actual number is
also divisible by 3.
For Example
Let’s take the number 435 and sum its digits up to check whether it is
divisible by 3 or not. The sum of digits is 4 + 3 + 5 = 12, and 12 is
completely divisible by 3. Therefore, 435 is divisible by 3.

4. Let’s take another example to understand the rule of d ivisibility by 3 in
depth. Consider a number 506 and take its sum which is 5 + 0 + 6 = 11. As
11 is not the multiple of 3, it is not divisible by 3 either. Thus, this proves
that 506 is not divisible by 3.
Rule No. 4: Divisibility by 4
According to the divisibility rule of 4, if the last two digits of a specific
number are divisible by 4 or the multiple of 4, that number is completely
divisible by 4. Moreover, if the last two digits are 00, the respective
number is also wholly divisible by 4.
For Example
To check whether 4920 is divisible by 4 or not, take the last two digits,
i.e., 20, and if they are a multiple of 4, then the number is divisible by 4
completely. Here 20 is divisible by 4, and so 4920 is.
Rule No. 5: Divisibility by 5
Most easy to remember the rule. If the number ends with digit 0 or 5,
then it is completely divisible by 5.
For Example
Any number ending with 0 or 5 such as 105, 800, 165000, 6934580,
656005, etc.
Rule No. 6: Divisibility of 6
The rule of divisibility by 6 states that a number is completely divisible by
6 if they are divisible by 2 and 3. If the sum of the digits of a given
number is multiple of 3 and the last digit is even, then it is concluded that
the number is the multiple of 6.

5. For Example
Check if 530 is divisible by 6 or not?
The last digit is 0, which is divisible by 2. Now sum the digits, 5 + 3 + 0 =
8, which is not divisible by 3. Hence, 530 is not divisible by 6.
Check if 360 is divisible by 6 or not?
The last digit is 0, which is divisible by 2. Now sum the digits, 3 + 6 + 0 =
9, which is completely divisible by 3. Hence, 360 is divisible by 6.
Rule No. 7: Divisibility of 7
Here comes one of the complex divisibility rules of mathematics. Follow
the steps given below to understand the divisibility rule of 7:
• Take the last digit of the given number
• Subtract twice of it from the remaining number
• Repeat the process till you get the two -digit number (if needed)
• Check if the remaining number 2-digit is the multiple of 7 or not.
• If it is, then the given number is completely divisible by 7
For Example
Is 905 divisible by 7?
Follow the procedure of divisibility rule of 7 step -by-step:
• Take the last digit 5 and double it, which becomes 10.
• Subtract it from the remaining number, 90 – 10 = 80

6. • As 80 is not a multiple of 7. Therefore, 905 is not completely
divisible by 7.
Is 37961 divisible by 7?
Follow the procedure of divisibility rule of 7 step -by-step:
• Take the last digit 1 and double it, wh ich becomes 2
• Subtract it from the remaining number, 3796 – 2 = 3794
• Still, a large number repeat the process.
• Take the last digit 4 and double it, which becomes 8
• Subtract it from the remaining number, 379 – 8 = 371
• Still, it is a large number. Repeat the process to get a two-digit
number
• Take the last digit 1 and double it, which becomes 2
• Subtract it from the remaining number, 37 – 2 = 35
• Now, 35 is a multiple of 7
Hence, the number 37961 is completely divisible by 7.
Rule No. 8: Divisibility of 8
This rule states that a number is divisible by 8. If the last three digits of
the given number are completely divisible by 8, then the given number is
completely divisible by 8. Moreover, if a number ends with 000, it is also
divisible by 8.
For Example
Take two numbers, 901816 and 675302, and determine if they are divisible
by 8 or not.

7. → First Number = 901816
Take its last 3 digits = 816, which is divisible by 8
Hence, 901816 is completely divisible by 8.
→ Second Number = 675302
Take its last 3 digits = 302, which is not divisible by 8
Hence, 675302 is not completely divisible by 8.
Rule No. 9: Divisibility by 9
Recall the divisibility rule of 3 because it is similar to that. According to
the divisibility rule of 9, a number is completely divisible by 9 if the sum
of its digits is the multiple of 9 or divisible by 9.
For Example
The number is 2169.
First, calculate the sum of its digits as 2 + 1 + 6 + 9 which is 18.
The sum of the digits is divisible by 9.
Hence, the number 2169 is completely divisible b y 9.
Rule No. 10: Divisibility by 10
Just like the divisibility rule of 5 but this time, if a number ends with 0
digits, then it is completely divisible by 10. In other words, any number
with 0 at its ones’ place digit is divisible by 10.
For Example
1560, 10000, 50, 5400, and many numbers with zero at ones’ place are
completely divisible by 10.
Rule No. 11: Divisibility by 11

8. This divisibility rule states that a number is only completely divisible by
11 if the difference of the sums of its alternative digi t of a given number
is divisible by 11.
Let us go through the procedure of adding alternative digits of a number
to understand the divisibility rule of 11:
• Consider a number 1364
• First, group its alternative digit, which is 16 and 34
• Take the sum of each group digit now, i.e., 1 + 6 and 3 + 4, which
are 7 and 7, respectively.
• Now calculate the difference of the sum like 7 – 7, which is 0
• Here 0 is the difference which is divisible by 11.
Therefore, 1364 is completely divisible by 11
Additional Rules of Divisibility by 11
Odd Number of Digits Rule:
If the number of digits of the given number is odd, like 3, 5, 7, etc., then
subtract the first and last digits from the remaining numbers. This way,
you will get the multiple of 11, which p roves that the given number is
completely divisible by 11.
Even Number of Digits Rule:
suppose the number of the digits of a given number is even like 2, 4, 6,
etc. In that case, adding the first digit and subtracting the last digit from
the remaining number will give you the multiple of 11, concluding that the
given number is completely divisible by 11.
Group of Two Digits Rule:

9. Split the number into two groups and then take the sum of the resultant
groups. Take the right end digits and left end digits to make groups. The
given number should be divisible by 11 if the sum of groups is a multiple
of 11.
Minus the Digit Rule:
Another procedure to determine if the number is divisible by 11, subtract
the last digit from the remaining number. Repeat the process u ntil you
get the two-digit number. If the resultant two -digit value is a multiple of
11, then the given number will be completely divisible by 11.
Rule No. 12: Divisibility by 12
The divisibility rule of 12 states that if a number is completely divisible b y
3 and 4, it is also divisible by 12.
For Example
Consider number 648
→ 648/3 = 216 (divisible by 3)
→ 648/4 = 162 (divisible by 4)
Therefore, the number 648 is completely divisible by 12.
Rule No. 13: Divisibility by 13
13 is a complex number, and so the ir divisibility rule is. According to the
divisibility rules, to determine if a number is completely divisible by 13,
calculate 4 times the last digit and add it to the remaining number. If the
answer is not a two-digit number, repeat the process until you get it.
Once you get the two-digit number, check whether it is divisible by 13 or
not. If the calculated two-digit number is completely divisible by 13 or
multiple of 13, then the given number is divisible by 13.

10. For Example
Consider the number 156 and multiply its last digit four times.
156 → 6 x 4 = 24
Now add 24 into the remaining number
→ 15 + 24 = 39
According to the multiples of 13,
→ 13 x 3 = 39
Thus, the number 156 is completely divisible by 13.
Rule No. 14: Divisibility by 14
There are two rules to determine whether a number is completely
divisible by 14 or not. First, if the given number is completely divisible by
2 and 7, it is also divisible by 14. The second rule to identify the
divisibility of a number by 14 is by adding the last two digits tw ice to the
remaining number.
For Example
Consider a number 1764
Check whether it is divisible by 2 and 7
→ 1764/2 = 882
→ 1764/7 = 252
It shows that the given number is divisible by 2 and 7.
Hence, 1764 is completely divisible by 14.
Rule No. 15: Divisibility by 15
According to the divisibility rule of 15, if the sum of the digit of given
numbers is divisible by 3, then the given number is completely divisible by
15.
For Example

11. The number is 84963325
Now, 8 + 4 + 9 + 6 + 3 + 3 + 2 + 5 = 40
Thus, 40 is not divisible by 3
Hence, 84963325 is not completely divisible by 15.
Rule No. 16: Divisibility by 16
The divisibility rule of 16 states that if the thousands of digits of the given
number is even and the last three digits are divisible by 16, then the given
number is also divisible by 16 completely. Another way to determine a
number is divisible by 16 is by adding 8 in the last three digits if the
thousands digit is odd.
For Example
126,320 is a number with an even number at its thousand digits, i.e., 6.
Check the last 3 digits divisibility by 16. Therefore, 126,320 is completely
divisible by 16.
Another example with an odd number at thousands of digits. Take 223,497
and add 8 into its last three digits, such as 497 + 8 = which is divisible by
16. Thus, the given number 223,497 is completely divisible by 16.
Rule No. 17: Divisibility by 17
Multiply the last digit of the given number by 5 and subtract it from the
remaining number. If the resultant number is divisible by 17, then the
given number is completely divisible by 17.
For Example
Consider the number 986 and subtract 30 from 98.
Here 30 is obtained by multiplying 6 with 5, 6 x 5 = 30.

12. Now, the resultant value is 68, which is divisible by 17.
Hence, the given number 986 is completely divisible by 17.
Rule No. 18: Divisibility by 18
According to the divisibility rule of 18, a number is completely divisible by
18 if the sum of its digit is divisible by 9. Moreover, the number should be
even; however, it is not always compulsory.
For Example
Check whether 7110 is divisible by 18 or not?
Take sum of digits, 7 + 1 + 1 + 0 = 9
Since 9 is the multiple of 9
Therefore, 7110 is completely divisible by 18.
Rule No. 19: Divisibility by 19
This rule states that to determine the divisibility of a number by 19,
multiply the last digit with 2, then add the product result in the remaining
number. If the resultant value is divisible by 19, then the actual number is
also divisible by 19 completely.
For Example
Check whether 1235 is divisible by 19 or not.
By divisibility rule of 19, 123 + (5 x 2) = 133, and 133 is divisible by 19.
Hence, the given number 1235 is completely divisible by 19.
Rule No. 20: Divisibility by 20
The divisibility rule of 20 has two conditions, and each should be met to
get zero remainder. A number is completely divi sible by 20 if 0 is at its
ones’ digit place and any even number at its tens’ place.

13. For Example
345,460 is completely divisible by 20 because it has 0 at its one digit and
an even number (6) at its ten digits.
Frequently Asked Questions
What are divisibility tests?
The divisibility test is a method to determine if the given number is
completely divisible by a particular divisor without leaving a reminder.
When a number or integer is completely divided by another number, the
remainder is zero, and a quotient is a whole number.
Write down the divisibility rule for 6 with an example.
The rule of divisibility by 6 states; if the actual number is divisible by both
2 and 3 and the sum of digits is a multiple of 3, then that number is
completely divisible by 6.
For example, the number 114 is completely divisible by 6 since it is even,
and the sum of digits is the multiple of 3.
Why are divisibility rules important?
Math is not a piece of cake for everyone. Even the experts follow the
rules, quick tips, and shortcuts to solve complex and lengthy problems.
Divisibility rules are the immediate methods to determine whether a
number is completely divisible by a specific number or integer.
Learning divisibility rules is essential as they are handy to solve word
problems, perform quick calculations and check prime numbers.
Give an example of the divisibility rule of 9?
9162 is a perfect example of disability rule by 9 as the sum of its digits is
also divisible by 9, which is 18.

14. Check whether 100244 is divisible by 4 or not?
To check 100244 divisibility by 4, consider the last two digits, i.e., 44,
which is divisible by 4. Therefore, the given number 100244 is completely
divisible by 4.