Divisibility rule of 3 or Divisibilty Test of 3 is a simple mathematical guideline used to determine whether a given integer is divisible by 3 without performing the actual division operation.

Mathematics often presents us with clever shortcuts and tricks that make calculations faster and more intuitive. One such fascinating shortcut is the divisibility rule of 3. Imagine you have a large number, and you need to know if it’s divisible by 3 without reaching for a calculator. The divisibility rule of 3 comes to the rescue! This simple yet powerful rule allows you to quickly determine divisibility just by examining the sum of the digits. Intrigued? Let’s dive into how this rule works and why it’s a handy tool in your mathematical toolkit.

In this article, we will learn what is divisibility rule of 3 is, how to use divisibility rule of 3 with examples and solve some questions based on it.

What is Divisibility Rule?

A divisibility rule is a mathematical guideline used to determine whether one integer is divisible by another without performing the actual division operation.

Divisibility rules typically exist for specific divisors, such as 2, 3, 4, 5, 6, 9, 10, etc.

For example:

• The divisibility rule for 2 states that if the last digit of a number is even (0, 2, 4, 6, or 8), then the number is divisible by 2.

• The divisibility rule for 5 states that if the last digit of a number is 0 or 5, then the number is divisible by 5.

What is Divisibility Rule of 3?

Divisibility rule for 3 is a method to determine whether a given number is divisible by 3 without actually performing the division. The rule states:

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

This rule simplifies the process of checking whether a number is divisible by 3. By this we can quickly determine the divisibility without performing long division. This rule is particularly useful in mental math, simplifying fractions, and various mathematical applications.

Divisibility Rules of 3 with Examples

To determine a number is divisible by 3 we will follow the below given steps:

• Sum the Digits: Add up all the digits of the number.

• Check the Sum: Determine if the resulting sum is divisible by 3.

• Conclusion: If the sum is divisible by 3, then the original number is divisible by 3; otherwise, it is not.

Now let us consider few examples for better understanding:

Example 1: Check if 813 is divisible by 3 or not.

Solution:

• Sum of digits: 8 + 1 + 3 = 12
• Check the sum: 12 is divisible by 3.
• Conclusion: 12 is divisible by 3, so 813 is divisible by 3.

Example 2: Check if 932 is divisible by 3 or not.

Solution:

Given number is 932

• Sum the digits: 9 + 3 + 2 = 14
• Check the sum: 14 is not divisible by 3.
• Conclusion: 932 is not divisible by 3

Divisibility Rule of 3 for Large Numbers

The divisibility rule of 3 for large numbers follows the same principle as for smaller numbers. Divisibility Rule of 3 for large numbers is an arithmetic shortcut that helps determine whether a given large number is divisible by 3 without performing the actual division.

Procedure for Large Numbers:

• Sum the Digits: Add up all the digits of the given number.

• Check the Sum: Determine if the sum obtained in step 1 is divisible by 3.

• Conclusion: If the sum is divisible by 3, then the original number is also divisible by 3; otherwise, it is not.

Let’s understand this with an example:

Example: Consider the large number 123456789123456789: Check its divisibility by 3 without performing actual division.

Solution:

• Sum the Digits: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 99
• Check the Sum: 99 is divisible by 3.
• Conclusion: Therefore, the large number 123456789123456789 is divisible by 3.

Divisibility Rule of 3 and 9

The divisibility rule for 3 helps determine if a number is divisible by 3 without performing division. Here is the detailed process:

• Sum the Digits: Add all the digits of the number together.

• Check the Sum: If the resulting sum is divisible by 3, then the original number is also divisible by 3.

The divisibility rule for 9 is similar to the rule for 3. Here is the detailed process:

• Sum the Digits: Add all the digits of the number together.

• Check the Sum: If the resulting sum is divisible by 9, then the original number is also divisible by 9.

Let us take an example for the same:

Example: Check if the number 456 divisible by 3 and 9?

Solution:

Divisibility by 3:

• Sum the digits: 4 + 5 + 6 = 15
• Check the sum: 15 is divisible by 3.
• Hence, 456 is divisible by 3.

Divisibility by 9:

• Sum the digits: 4 + 5 + 6 = 15
• Check the sum: 15 is not divisible by 9.
• Hence, 456 is not divisible by 9.

Therefore, 456 is divisible by 3 but not divisible by 9.

Divisibility Rule of 3 and 6

As we have already seen the divisibility rule of 3 is determined by the sum of digits of the given number. If the sum obtained is divisible by 3 then the number is divisible by 3. Now let us see the divisibility rule of 6:

Divisibility Rule of 6

The divisibility rule for 6 is based on the rules for both 2 and 3. A number is divisible by 6 if it is divisible by both 2 and 3.

• Check for 2: If the last digit of the number is even (0, 2, 4, 6, or 8), then the number is divisible by 2.
• Check for 3: Apply the rule for divisibility by 3 (sum of digits divisible by 3).

Example: Check if the number 756 divisible by 3 and 6?

Solution:

Divisibility by 3:

• Sum the digits: 7 + 5 + 6 = 18
• Check the sum: 18 is divisible by 3.
• Hence, 756 is divisible by 3.

Divisibility by 6:

• Check for 2: Last digit is even (6), so divisible by 2.
• Check for 3: We have already seen 756 is divisible by 3.
• Hence, 756 is divisible by 6

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