Divisibility Rules: Complete Guide with Examples

Divisibility rules are a set of rules that can be used to determine if a number is evenly divisible by another without actually dividing it. These rules help us to quickly find solutions to math problems without following lengthy procedures. Learning how to apply these rules will make calculations fun and easier for you. By going through this article you will learn both basic( from1 to 10) and advanced(11 to 20) divisibilty rules applied to solve math problems. With the sample problems provided on this page you can grasp the concept easily.

 

Table of Contents

• What Is Divisibility?
• What Are Divisibility Rules?
• Divisibility Rules Chart (1 to 20)
• Basic Divisibility Rules (1 to 10)
• Advanced Divisibility Rules (11 to 20)
• Divisibility Rules Examples
• Misconceptions About Divisibility Rules
• Fun Facts
• Solved Examples
• Conclusion
• FAQs on Divisibility Rules

 

What Is Divisibility?

Divisibility is a property of a number to be divided by another number without leaving a remainder. For example, if a number is divided evenly by another number, without remainder-it is divisible by that number. For example, if you divide 20 with 5 and get remainder 0, then 20 is divisible by 5. Divisibility rules are shortcuts that help you figure out very quickly whether one number can be divided by another number. Let's understand in detail about the rules for divisibility.

What Are Divisibility Rules?

Divisibility rules are shortcut techniques used in mathematics to decide whether one quantity can be divided by another without performing full division. These rules help you quickly check if a number is divisible by using simple methods, such as analysing the digits or their sum. For instance, if the sum of a variety of digits is divisible by 3, then the entire number is divisible with the aid of 3. These rules are especially beneficial in checks, aggressive checks, intellectual math, and real-world conditions in which brief calculations are required. Understanding what's divisibility regulations can significantly improve your quantitative feel and velocity in solving problems.

 

Divisibility Rules Chart (1 to 20)

 

 

Basic Divisibility Rules (1 to 10)

Divisibility by  1

Each number is divisible by 1

There’s no unique rule required.

 

Divisibility by 2

A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.

Example: Is 128 divisible by 2?

Yes! It ends in 8, so 128 is divisible by 2.

 

Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Example:  Is 123 divisible by 3? 

Add the digits: 1 + 2 + 3 = 6 (which is divisible by 3).  

Therefore, 123 is divisible by 3.

 

Divisibility by 4

Look at the last 2 digits.

If they are divisible by four, then the number is divisble by 4.

Example: 

Is 212 divisible by 4? 

Yes! The last digit is 12, so 212 is divisible by 4.

 

Divisibility by 5

A number is divisible by 5 if its last digit is 0 or 5.

Example: 

Is 550 divisible by 5? 

Yes! The last digit is 0, so 550 is divisible by 5.

 

Divisibility by 6

A number is divisible by 6 if it is divisible by both 2 and 3.

Example: 

Is 36 divisible by 6?

36 ÷ 2 = 18 (divisible by 2)

Adding the digits: 3 + 6 = 9 (divisible by 3)

Since 36 is divisible by both 2 and 3, it is also divisible by 6.

 

Divisibility by  7

Double the final digit and subtract it from the rest.

If the result is divisible by way of 7, so is the number.

 

Divisibility by 8

Check the last three digits.

If they are divisible by 8, then the number is divisible too.

 

Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

Example:

Is 234 divisible by 9?  

Add the digits: 2 + 3 + 4 = 9, which is divisible by 9.  

Therefore, 234 is divisible by 9.

 

Divisibility by 10

A number is divisible by 10 if its last digit is 0.

Example: 

Is 790 divisible by 10?

The last digit is 0, so 790 is divisible by 10.

 

Advanced Divisibility Rules (11 to 20)

Divisibility by 11

Add the sum of alternative digits and subtract them. If this difference is divisible by eleven, then so is the number.

Example:  Is 2728 divisible by 11?

Check for the digits on the 1st and 3rd place: 2 + 2 = 4

Even (2nd and 4th place): 7 + 8 = 15

Diff: 15 - 4 = 11 which is divisible by 11

Divisibility by 12

A number is divisible by 12 if it is divisible by both 3 and 4.

Example:  Is 144 divisible by 12?  

144 ÷ 3 = 48 (divisible by 3)  

The last two digits are 44, and 44 ÷ 4 = 11 (divisible by 4).  

Since 144 is divisible by both 3 and 4, it is also divisible by 12.

Divisibility by 13

Add 4 times of the last digit to the rest of the number digits and repeat until you get a 2-digit number.

If that 2 digit number is divisible by 13 then the number is divisible by 13 else not. 

Example: Is 3763 divisible by 13?

The last digit of the given number is 7. After multiplying it with 4 we get, 28

Divisibility by 14

Must fulfils the divisibility rules of 2 and 7.

Example: Is 448 Divisible by 14?

1. Check is 448 divisible by 2, yes it is an even number.

2. Check is 448 divisible by 7: 448 ÷ 7 = 64

Yes, 448 is divisible by 14 as it fulfil divisibility rules of 2 and 7.

 

Divisibility by 15

Must fulfil divisibility by using 3 and 5.

Example: Is 345 divisibile by 15?

1. 445 is divisible by 5 (ends with 5).

2. Sum of digits = 3 + 4 + 5 =12, divisible by 3.

So, 345 is divisible by 15.

Divisibility by 16

The last four digits must be divisible by 16.

Example: Is 7,152 divisible by 16?

Last 4 digits = 7152

7152÷16 = 447

Yes, 7,152 is divisible by 16.

Divisibility by 18

Must be divisible by 2 and 9.

 

Divisibility by 19

Add two times the last digit to the rest and repeat the method.

 

Divisibility by 20

Ends with 0, and the quantity before the last must be even.

Example: Is 1020 divisible by 20?

1020 ends in 0, the second-final digit is even (2) 

Yes, 1020 is divisible by 20

 

Divisibility Rules Examples

Example 1: Is 462 divisible by 3?

Let's find the sum of all three digits of 462

4+6+2 = 12,

12 is divisible by 3

Yes, 462 is divisible by 3.

 

Example 2: Is 1,248 divisible by 4?

The last two digits of 1,284 is 48 which is divisible by 4.

Yes, 1,248 divisible by 4.

 

Example 3: Is 3355 divisible last means of 5?

Yes, 3355 ends in 5 so it is divisible with 5.

 

Example 4: Is 1441 divisible by 11?

We know that the divisibilty rule of 11 states that if the sum of digits on odd place−sum of digits on even places digits is divisible by 1 then the number is divisible by11.

Digits in odd place are 1 and 4

And, digits in even place are 1 and 4  

Therefore, difference of the sum = 1−4 + 4−1 = 0.

Yes, 1441 is divisible by 11

 

Example 5: Is 1020 divisible by 20?

Ends in 0, the second-last digit is even (2)  Yes

 

Misconceptions About Divisibility Rules

All even numbers are divisible by 4.

False: Only if the last digits are divisible by 4.

 

If a number is divisible by 3, it is divisible by 6.

False: It should additionally be even.

 

All numbers ending in 5 are divisible by way of 10.

False: Must end in 0 to be divisible by 10.

 

Rules may be used for decimals.

False: Divisibility guidelines work most effectively on complete numbers, no longer decimals.

 

Divisibility by 7 and effectively critical.

False: Very useful in exams and puzzles.

 

Fun Facts

Barcodes & ISBN Numbers

Use divisibility policies for error-checking the usage of mod 10 or 11.

 

Digital Puzzles & Aptitude Tests

Many puzzles in competitive assessments check divisibility logic.

 

Banking & Finance

Used in validating massive batches of numbers, like account digits.

 

Programming & Algorithms

Divisibility assessments assist in writing optimized codes.

 

Cryptography

Modular arithmetic is predicated closely on these policies for encoding.

 

Solved Examples

Example 1: Is 231 divisible by 3?

2+3+1 = 6, divisible by 3 Yes

 

Example 2: Is 1274 divisible by using 2 and 7?

The last digit, four  divisible by 2

Apply 7-rule: 127 - 2×4 = 127-8= 119  divisible with the aid of 7  Yes

 

Example 3: Is 4620 divisible by way of 5 and 3?

Ends in 0  divisible with the aid of 5

Sum = 4+6+2+0 = 12  divisible using 3. Yes

 

Example 4: Is 7896 divisible by using four?

Last  digits = 96  96 ÷ 4 = 24  Yes

 

Example 5: Is 123456789 divisible by way of 9?

Sum = 45  divisible by 9  Yes

 

Conclusion

Learning divisibility of numbers using divisibilty rules are shortcuts in math. Whether you're simplifying numbers, getting ready for exams, or solving puzzles, these values are highly useful. So, mastering divisibility rules is a step toward learning arithmetic!

 

Frequently Asked Questions on Divisibility Rules

1. What are the divisibility rules of 2 to 11?

Answer:

• 2= A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).

• 3= Add the digits. If the sum is divisible by 3, so is the number.

• 4= Check the last two digits. If they form a number divisible by 4, the entire number is.

• 5= If the number ends in 0 or 5, it's divisible by 5.

• 6=If a number is divisible by both 2 and 3, it’s divisible by 6.

• 7=Double the last digit and subtract it from the rest. If the result is divisible by 7, so is the number.

• 8=If the last three digits form a number divisible by 8, so is the whole number.

• 9=If the sum of the digits is divisible by 9, the number is divisible by 9.

• 10=If the number ends in 0, it is divisible by 10

• 11=Subtract and add digits alternately. If the alternating sum is divisible by 11 (or equals 0), the number is divisible by 11.

 

2. What is the divisibility rule of 11111?

Answer: There is no standard divisibility rule for 11111 like there is for small numbers. To check if a number is divisible by 11111, divide it directly or use factoring (if possible). 11111 is a prime number, so it's only divisible by 1 and itself unless part of a specific pattern.

 

3. What is divisible by 101?

Answer: Any number that is a multiple of 101 (e.g., 101, 202, 303, 404...) is divisible by 101.
To test if a large number is divisible by 101, you generally divide it directly since there is no common divisibility shortcut for 101.

 

4. How many integers are divisible by 1?

Answer: Every integer is divisible by 1.
No test or calculation is needed, any whole number divided by 1 equals itself, with no remainder.

 

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