# Divisibility Rule of 3

Created by: Team Maths - Examples.com, Last Updated: June 10, 2024

## Divisibility Rule of 3

Divisibility rules, particularly for Numbers like 3, offer a quick way to determine if one number is divisible by another without extensive computation. This concept intersects various areas of mathematics including Algebra, where it assists in simplifying expressions, where it aids in data categorization. Understanding these rules enhances comprehension of broader mathematical concepts such as Integers, Rational and Irrational Numbers, Additionally, methods to utilize these fundamentals to optimize fits in data analysis, showcasing the pervasive application of basic arithmetic principles across advanced mathematical topics.

## What is the Divisibility Rule of 3?

The divisibility rule for 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3. This simple check allows for quick assessments of divisibility, useful in various mathematical calculations and problem-solving scenarios. For example, the number 12345 is divisible by 3 because the sum of its digits (1+2+3+4+5=15) is divisible by 3.

## Proof of Divisibility Rule of 3

Let’s use the number 2763 to prove the divisibility rule of 3 with a similar step-by-step example.

Step 1: Consider the number 2763.

Step 2: Let’s expand this number as shown below:

Step 3: 2763 = 2 × 1000 + 7 × 100 + 6 × 10 + 3 × 1

= 2 × (999 + 1) + 7 × (99 + 1) + 6 × (9 + 1) + 3 × 1

= (2 × 999 + 7 × 99 + 6 × 9) + (2 × 1 + 7 × 1 + 6 × 1 + 3 × 1)

= (2 × 999 + 7 × 99 + 6 × 9) + (2 + 7 + 6 + 3)

Step 4: We know that numbers 9, 99, 999, etc., are divisible by 3, making their multiples also divisible by 3.

Step 5: Thus, the question of the divisibility of 2763 boils down to whether the sum 2 + 7 + 6 + 3 is divisible by 3.

Step 6: Here, 2, 7, 6, and 3 are the digits of the number 2763.

Step 7: Adding these digits gives us 2 + 7 + 6 + 3 = 18, which is clearly divisible by 3.

Step 8: From this example, we conclude that if the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3.

## Divisibility Rule of 3 and 9

### Divisibility Rule of 3

The divisibility rule for 3 is straightforward: a number is divisible by 3 if the sum of its digits is divisible by 3. This rule is particularly useful because it simplifies the task of checking divisibility without performing actual division. For example, the number 621 is divisible by 3 since the sum of its digits (6 + 2 + 1 = 9) is divisible by 3.

### Divisibility Rule of 9

Similarly, the divisibility rule for 9 is closely related to that of 3: a number is divisible by 9 if the sum of its digits is divisible by 9. This method checks for divisibility efficiently and quickly. For instance, consider the number 7386; adding its digits gives 7 + 3 + 8 + 6 = 24. Since 24 is not divisible by 9, 7386 is also not divisible by 9.

## Divisibility Test of 3 and 4

The divisibility test for 3 is simple: a number is divisible by 3 if the sum of its digits is divisible by 3. This test allows you to quickly determine whether a number can be divided by 3 without performing the division operation.

### Example:

• Number: 294
• Digits Sum: 2 + 9 + 4 = 15
• Check: 15 is divisible by 3 (15 ÷ 3 = 5).
• Conclusion: 294 is divisible by 3.

## Divisibility Rule of 3 Examples

### Number: 123

• Digits Sum: 1 + 2 + 3 = 6
• Is 6 divisible by 3? Yes
• Conclusion: 123 is divisible by 3.

### Number: 456

• Digits Sum: 4 + 5 + 6 = 15
• Is 15 divisible by 3? Yes
• Conclusion: 456 is divisible by 3.

### Number: 569

• Digits Sum: 5 + 6 + 9 = 20
• Is 20 divisible by 3? No
• Conclusion: 569 is not divisible by 3.

### Number: 981

• Digits Sum: 9 + 8 + 1 = 18
• Is 18 divisible by 3? Yes
• Conclusion: 981 is divisible by 3.

### Number: 2345

• Digits Sum: 2 + 3 + 4 + 5 = 14
• Is 14 divisible by 3? No
• Conclusion: 2345 is not divisible by 3.

## How do you apply the divisibility rule of 3 to a number like 276?

Add the digits: 2 + 7 + 6 = 15. Since 15 is divisible by 3, 276 is also divisible by 3.

## Is there an example of a number that is not divisible by 3?

Consider the number 214: 2 + 1 + 4 = 7. Since 7 is not divisible by 3, neither is 214.

## Can the divisibility rule of 3 be applied to very large numbers?

Yes, the rule applies to any number, no matter how large, as long as you can sum its digits to check divisibility by 3.

## Does the divisibility rule of 3 work with negative numbers?

Yes, it does. For instance, -123 has a digits sum of 1 + 2 + 3 = 6, which is divisible by 3, so -123 is divisible by 3.

## How is the divisibility rule of 3 useful in everyday mathematics?

It’s useful for simplifying fractions, checking if numbers are factors of larger numbers, and in division problems to avoid detailed calculations.

## What should I do if the sum of the digits is also a large number?

If the sum is large, continue to add the digits of the sum until you get a smaller number that can easily be evaluated for divisibility by 3.

## Are there any exceptions to the divisibility rule of 3?

No, the rule is universally applicable to all integers as long as the correct process of summing the digits and checking for divisibility is followed.

## What is a quick example to demonstrate the rule?

For the number 591, add the digits: 5 + 9 + 1 = 15. Since 15 is divisible by 3, so is 591.

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