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.

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 whole 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.

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 article 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 quickly in-depth.

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 integers are divisible by 1.

Take any number like 5 or 6000, and they are completely divisible by 1.

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.

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.

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.

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.

Let’s take another example to understand the rule of divisibility 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.

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.

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.

Most easy to remember the rule. If the number ends with digit 0 or 5, then it is completely divisible by 5.

Any number ending with 0 or 5 such as 105, 800, 165000, 6934580, 656005, etc.

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.

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.

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

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
- 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, which 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.

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.

Take two numbers, 901816 and 675302, and determine if they are divisible by 8 or not.

→ 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.

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.

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 by 9.

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.

1560, 10000, 50, 5400, and many numbers with zero at ones’ place are completely divisible by 10.

This divisibility rule states that a number is only completely divisible by 11 if the difference of the sums of its alternative digit 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

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 proves that the given number is completely divisible by 11.

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.

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.

Another procedure to determine if the number is divisible by 11, subtract the last digit from the remaining number. Repeat the process until 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.

The divisibility rule of 12 states that if a number is completely divisible by 3 and 4, it is also divisible by 12.

Consider number 648

→ 648/3 = 216 (divisible by 3)

→ 648/4 = 162 (divisible by 4)

Therefore, the number 648 is completely divisible by 12.

13 is a complex number, and so their 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.

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.

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 twice to the remaining number.

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.

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.

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.

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.

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.

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.

Consider the number 986 and subtract 30 from 98.

Here 30 is obtained by multiplying 6 with 5, 6 x 5 = 30.

Now, the resultant value is 68, which is divisible by 17.

Hence, the given number 986 is completely divisible by 17.

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.

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.

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.

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.

The divisibility rule of 20 has two conditions, and each should be met to get zero remainder. A number is completely divisible by 20 if 0 is at its ones’ digit place and any even number at its tens’ place.

345,460 is completely divisible by 20 because it has 0 at its one digit and an even number (6) at its ten digits.

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.

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.

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.

9162 is a perfect example of disability rule by 9 as the sum of its digits is also divisible by 9, which is 18.

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.

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