Divisibility Rules

A divisibility rule is a simple shortcut that helps us check whether a number can be divided by another number by looking at its digits, without doing long division. These rules, also called divisibility tests, make large calculations easier and help us determine whether one number is divisible by another quickly.
 

Example:
 
Every number is divisible by 1.
 

\(7 ÷ 1 = 7\)
 

\(245 ÷ 1 = 245\)
 

So any whole number we pick will always be divisible by 1.

Divisibility rules help us quickly check whether a number is divisible by another number without performing long division. These rules make large calculations easier and help us simplify numbers during problem-solving.
Below is a table of commonly used divisibility rules:
 

Let’s check each number using the correct divisibility rule:
 

• Is 280 divisible by 2?
  Yes. The last digit is 0, which is even, so 280 is divisible by 2.
   
• Is 345 divisible by 3?
  Yes. Add the digits: 3 + 4 + 5 = 12, and 12 is divisible by 3. So, 345 is divisible by 3.
   
• Is 450 divisible by 4?
  No. Look at the last two digits: 50. Since 50 is not divisible by 4, 450 is not divisible by 4.
   
• Is 3900 divisible by 5?
  Yes. The last digit is 0, so the number is divisible by 5.

We already know the basic divisibility rules for 2, 3, 5, 7, and 11. Now let’s build on that and learn simple tests for prime numbers between 11 and 19, starting with 13 and 17.

Divisibility Rule for 13:

A number is divisible by 13 if it divides evenly, with no remainder. Here’s a quick method to test it without using long division:
 

• Take the last digit of the number.
   
• Multiply that digit by 4.
   
• Add the result to the rest of the number (ignore the last digit).
   

If the final answer is divisible by 13, then the original number is also divisible by 13. There are three additional methods to check divisibility by 13, but this is the simplest one for learners.

Divisibility Rule for 17:

A number is divisible by 17 if it is entirely divided without leaving any remainder. Here’s a shortcut to check it:
 

• Take the last digit of the number.
   
• Multiply that digit by 5.
   
• Subtract the result from the remaining part of the number.
   

If the final value is divisible by 17, then the original number is also divisible by 17.

Some tips and tricks for mastering in divisibility rules are given below. 

• Begin by familiarizing yourself with the divisibility rules for numbers from 1 to 10. This will help to give a strong foundation. 
   
• For divisibility by 3 or 9, add up all digits of the number. If the sum is divisible by 3 or 9, then the original number is divisible by that number as well. 
   
• For divisibility by 4, check if the number formed by the last two digits is divisible by 4. Similarly, for divisibility by 8, the last three digits should form a number divisible by 8. 
   
• To test divisibility by 11, subtract the sum of the digits in odd positions from the sum of the digits in even positions. If the result is divisible by 11, then the original number also is divisible by 11. 
   
• Consistently practice with various numbers, and it will help to improve speed and accuracy in calculating. 
   
• Use visual aids such as a divisibility rules chart, a PDF, or posters at home or in the classroom.
   
• Practice regularly with a Divisibility Rules worksheet or a Divisibility Rules worksheet PDF to build confidence.
   
• Teach simple shortcuts, such as checking the last digit or adding digits, before moving on to more complex rules.
   
• Use games or timed challenges to make learning fun and competitive.
   
• Show how the rules for divisibility help in the larger topics, like simplifying fractions or checking multiples.

Divisibility rules are used widely in various fields. Here are a few real-world applications of divisibility rules:

• Banking: Banks use divisibility rules to verify account and credit card numbers. They use divisibility rules to detect any kind of error in large transactions and account details.
  
   
• Scheduling: Schools and offices use divisibility rules to divide time efficiently.
  
   
• Packaging and distribution: To distribute products efficiently, companies use divisibility rules to ensure even distribution.
  
   
• Time management: For planning events and scheduling, time has to be managed by allocating time slots evenly. This can be done by dividing the hours in equal way. 
  
   
• Budgeting: Using divisibility rules, expenses can be evenly divided. For instance, the total group expense can be divided evenly while travelling or having food.