## Introduction to Divisibility

A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although divisibility tests exist for numbers in any radix or base, they are different. Imagine 3 friends trying to share 10 cookies. Each gets 3 cookies, and there’s one left over. They are unsure what to do with it; would one person get an extra cookie? That did not seem fair to the 3 friends, who loved sharing everything equally.

If there were 9 cookies, they would have divided the cookies equally, and there would have been no confusion. 9 is divisible by 3. This means that 9 cookies could have been divided into three equal parts without any extra cookies left. Further, we will study divisibility rules by applying their rules.

Divisibility Concept

## Divisibility Rules

This section will teach about basic divisibility tests from 2 to 8. The divisibility rule of 1 is not required since every number is divisible by 1. Here are a few basic divisibility rules:

## Divisibility Tests by 9 and 11

Divisibility test by 9: The sum of all the digits of the number should be divisible by 9.

Example: Using the divisibility rule of 9, state whether 724 is divisible by 9 or not.

Ans: Let us find the sum of all the digits of the number 724.

7+2+4 = 13

Here, 13 is not divisible by 9, so as per the divisibility test of 9, 724 is not divisible by 9.

Divisibility test by 11: The divisibility by 11 rule states that if the difference between the sum of the digits at odd places and the sum of the digits at even places of the number, is 0 or divisible by 11, then the given number is also divisible by 11.

Example: Test the divisibility of the 86416 by 11.

Ans: In 86416, if we take the alternate digits starting from the right, we get 6, 4, and 8 and the remaining alternate digits are 1 and 6. Now, 6 + 4 + 8 = 18, and 1 + 6 = 7. After finding the difference between these sums, we get 18 - 7 = 11, which is divisible by 11. Therefore 86416 is divisible by 11. It is to be noted that these alternate digits can also be considered as the digits on the odd places and the digits on the even places.

## Divisibility Rule of 25

Divisibility by 25 means that a number is divisible by 25. A number is divisible by 25 if the last or the final two digits of the number are divisible by 25. For example, 125 is divisible by 25 because the last two digits, 25, are divisible by 25.

When a number is divisible by 25, it means that the number can be divided evenly by 25. The number 50 is divisible by 25 because it can be evenly divided into two groups of 25. 100 is divisible by 25 because it can be evenly divided into four groups of 25.

Example: Check whether 5200 is divisible by 25.

Ans: According to the divisible rule of 25, if the last two digits of a number are zeroes or the number formed by the last two digits is a multiple of 25, then the number is divisible by 25.

In the given number 5200, the last two digits are zeroes.

So, the number 5200 is divisible by 25.

## Solved Examples

Q1. A number is divisible by 4 and 12. Check if it is divisible by 48.

Ans: 48 = 4 × 12 but 4 and 12 are not coprime.

No, it's not necessary that the number will be divisible by 48.

Q2. Salma wants to distribute 123 toffee equally among 15 of her friends. Use the rules of divisibility and check whether she will be able to do so.

Ans: 123 toffees distributed to 15 friends

$=1+2+3=6 \div 3=\text { Yes }(6 \text { is divisible by } 3)$

$=123 \div 5=\text { No }(123 \text { is not divisible by } 5)$

No, she cannot distribute 123 toffee equally.

Q3 Without actual division, find if 235948 is divisible by 4.

Ans: The number formed by the last two digits on the extreme right side of 235948 is 48

48 ÷ 4 = 12, i.e. 48 is divisible by 4.

Therefore, 235948 is divisible by 4.

## Practice Questions

Q1 Check whether 4410 is divisible by 45 or not.

Ans: Yes,4410 is divisible by 45.

Q2 State if the first number is divisible by the second number.

51 by 6

Ans: No, 51 is not divisible by 6.

Q3 Is 153 divisible by 9?

Ans: Yes

## Summary

Divisibility rules in math are a set of specific rules that apply to a number to check whether the given number is divisible by a particular number or not. In this article, we have seen the divisibility rules for different numbers. The divisibility rule of 9 says that the sum of the digits of the given number should be divisible by 9. However, the divisibility rule of 11 states that a number is divisible by 11 if the difference between the sum of the digits at even places and odd places is 0 or divisible by 11. Later on, we learned about the divisibility rule of 25. i.e. If the last or the final two digits of a number are divisible by 25, then the whole number is divisible by 25. In the end, we added some solved examples and practice problems to check the divisibility rules.

## FAQs on 25 Divisibility Rule

1. What is the use of divisibility rules?

In math, divisibility tests are important to learn as it helps us to ease out our calculations where we have to do multiplication and division. By applying divisibility rules, we can quickly identify whether a particular number is divisible by another number.

2. If a number is divisible by 8, will it be divisible by 4?

Yes, if a number is divisible by 8, it will also be divisible by 4 . 4 is the factor of 8. If a number ' $x$ ' is divisible by a number ' $y$ ', then ' $x$ ' will also be divisible by factors of ' $y$ '. Similarly, if a number is divisible by 8, it will be divisible by 4. The number will be divisible by 2 because 2 is also a factor of 8. So, if a number is divisible by 8, it will be divisible by 4 and 2.

3. How many divisibility rules are there?

Generally, we have divisibility rules from 1 to 20. But if we could identify the pattern of multiples of numbers, we could create more divisibility tests. For example, the divisibility rule of 21 states that a number must be divisible by both 3 and 7. It is because 21 is a multiple of two prime numbers, 3 and 7, so all the multiples of 21 will have 3 and 7 as their common factors.