Divisibility Rules

Divisibility Rules

In this mini-lesson, we will explore about divisibility rules by learning how to apply divisibility rules with examples, and the divisibility rules of specific numbers while discovering the interesting facts around them.

In a 1962 Scientific American article, the popular science writer, Martin Gardner, discusses divisibility rules for 2–12, where he explains that the rules were widely known during the renaissance and used to reduce fractions with large numbers down to the lowest terms.

Since every number is not completely divisible by every other number, they may leave a remainder other than zero. There are certain rules which help us determine the actual divisor of a number just by considering the digits of that number. These are called divisibility rules.

Do you think, 375 is divisible 6?

Example: Divisibility by 6

Let us all find answers to all such questions in this lesson on divisibility rules!

Lesson Plan


What Are Divisibility Rules?

A divisibility rule is a kind of shortcut to figure out if a given integer is divisible by a divisor, without performing the whole division process but by examining its digits.

Multiple divisibility rules can be applied to the same number which can quickly determine its prime factorization.

Divisibility rules are a set of general rules or heuristics that are used to figure out if a number is wholly divisible by another number.


What Are the Basic Divisibility Rules?

Here are a few basic divisibility rules:

Divisibility by number Divisibility Rule
Divisible by 2 A number is even or a number whose last digit is an even number i.e. 2, 4, 6, 8 including 0
Divisible by 3 The sum of the digits of the number is divisible by 3
Divisible by 4 The last two digits of the number are divisible by 4
Divisible by 5 A number which has 0 or 5 as their last digits
Divisible by 6 A number which 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 last three digits of a number are divisible by 8
Divisible by 9 The sum of the digits of the number is divisible by 9
Divisible by 10 Any number whose last digit is 0
Divisible by 11 The difference of the sum of the alternative digits of a number is divisible by 11
Divisible by 12 A number which is divisible by both 3 and 4

Examples

1)

Example: Divisibility by 6

2)

Example: Divisibility by 8

3)

Example: Divisibility by 11


What Are Intermediate Divisibility Rules?

Intermediate divisibility rules are applied to prime numbers which are less than 20 and greater than 10.

Divisibility by number Divisibility Rule
Divisibility by  13  If 4 times the units digit of the number plus the number obtained by removing the units digits of the number is a multiple of 13
      Divisibility by 17  If the units digit of a number is subtracted 5 times from the remaining (excluding the units digit) results in a number that is divisibly by 17        
 Divisibility by 19  If doubling the units digit and adding it to the number formed by removing the units digits in the original number is divisible by 19

Examples

1)

Example: Divisibility by 13

2)

Divisibility by 17

3)

Example: Divisibility by 19

 
Thinking out of the box
Think Tank
  1. A number is divisible by 4 and 12. Is it true that it will be divisible by 48?
  2. Check whether 2359334 follow divisibility rules of 4 as well as 8

How to Prove Divisibility Rules for 7 and 8?

Divisibility Rule for 7

Explanation

Step 1: Multiply the units digit by 2

Step 2: Subtract the product from the remaining digits.

Step 3: Check the difference.

If the difference is divisible by 7, then the number is divisible by 7

For example: Is 623 divisible by 7?

Step 1: 3 x 2 = 6
Step 2: 62 – 6 = 56
Step 3: The difference is 56 and 56 is divisible by 7
Thus, 623 is divisible by 7

Divisibility Rule for 8

Explanation

A number is divisible by 8 when the last three digits are evenly divisible by 8

For example, check if 17224 is divisible by 8

The last three digits are 224 and the number 224 is divisible by 8. Hence, 17224 is divisible by 8

 
important notes to remember
Important Notes
  1. Divisibility rules are of great importance while checking prime numbers.
  2. These are handy to solve word problems.
  3. They are useful to do quick calculations.

Solved Examples

Now let's go through some divisibility rules with examples.

Example 1

 

 

Khushi prepared 78 desserts. She arranged 3 desserts on each platter, with the same number of desserts on each platter.

How many platters did Khushi prepare?

Solved Example: Desserts on platters

Solution

Total desserts prepared = 78

We know that she arranged the same number of desserts on each platter.

Number of desserts on each platter = 3

Sum of the digits = 7 + 8 = 15

15 is divisible by 3

Thus, we can say that 78 is divisible by 3

Now, by dividing 78 by 3, we get 26

Thus, Khushi prepared 26 platters.

\(\therefore\) Khushi prepared 26 platters.
Example 2

 

 

Sahil collected 156 cans for an Art project. Kiara also collected a few cans. Together the total number of cans represent a number that is not divisible by 2

Which of the following represents the number of cans they collected together?

Solved Example: To find the number of cans

Select an option: 236, 254, 289, 278

Solution

A number is divisible by 2, if it is even or if the last digit is an even number, i.e. 2, 4, 6, 8 including 0

Thus, out of the listed numbers 236, 254, 289 and 278, only 289 is not divisible by 2

This indicates that they had collected 289 cans together.

\(\therefore\) They collected 289 cans together.

Example 3

 

 

Mr. Nakul sells tickets for film festivals.

The table below shows the number of tickets he sold for 2 days last month.

Which of the following statements is true?

Days Number of tickets sold
Monday  396
Tuesday  169

(a) The numbers of tickets sold on Tuesday is divisible by 13

(b) The number of tickets sold on Monday is divisible by 3, 6, and 9

Solution

(a) Using the divisibility rules of 13, we can say that 16 + (9x4) = 52 is divisible by 13. Therefore, the number of tickets sold on Tuesday is divisible by 13

(b) Yes. 396 is even and 3 + 9 + 6 = 18, thus divisible by 3. Hence, it is divisible by 6. Thus, the number is divisible by 9

\(\therefore\) Both the statements are true.
Example 4

 

 

Alia wants to find out whether 4848 is divisible by 8. Can you help her?

Solution

The last 3 digits 848 are divisible by 8

Hence, 4848 is divisible by 8

\(\therefore\) 4848 is divisible by 8.
Example 5

 

 

Kartik is arranging some photographs for his family book. He needs to put the same number of photographs in each row with no photographs left over.

If he has 288 photographs to arrange, how many photographs can he place in each row? Is it 5, 9, or 10?

Solution

Checking its divisibility by 5, 9, and 10, we can say that 288 is divisible by 9

Hence, 9 photographs can be placed in each row.

\(\therefore\) 9 photographs can be placed in each row.

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the 'Check Answer' button to see the result.

 
 
 
 
 
 

Let's Summarize

We hope you enjoyed learning about the Divisibility Rules with solved examples and practice questions. Now you will be able to easily solve problems on divisibility rules and their applications.

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Frequently Asked Questions (FAQs)

1. What is the divisibility rule for 36?

Check for divisibility by 4 and 9 to see if it is divisible by 36

2. What is the divisibility rule for 2?

If a number is even or a number whose last digit is an even number i.e. 2, 4, 6, 8 including 0, it is always completely divisible by 2

3. What is the 11 divisibility rule?

Subtract the last digit from a number made by the other digits. If that number is divisible by 11, then the original number is divisible by 11 too.

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