Divisibility Rule of 6 - Methods, Examples

What is the Divisibility Rule of 6?

The divisibility rule of 6 says that a number is divisible by 6 only if it is divisible by both 2 and 3. This means the number should meet two conditions:

1. It must be divisible by 2 – In simple terms, the number should be an even number. That means its last digit must be 0, 2, 4, 6, or 8.

2. It must be divisible by 3 – To check this, you just add up all the digits of the number. If the sum is divisible by 3, then the number passes the second test.

If a number passes both these tests, it is divisible by 6!

For example, consider the number 54. It ends in 4, so it is even and divisible by 2. The sum of its digits is 5 plus 4, which is 9, and 9 is divisible by 3. Since 54 is divisible by both 2 and 3, it is divisible by 6.

Read More: How to Do Long Division

Divisibility by 6 for Large Numbers

The divisibility rule of 6 is the same whether the number is small or large. To check if a large number is divisible by 6, we need to make sure it satisfies two conditions. 

• First, it must be divisible by 2. That means the number should be even, and you can check this by looking at the last digit. If the last digit is 0, 2, 4, 6, or 8, then the number is even and divisible by 2. 

• Second, the number must also be divisible by 3. This can be checked by adding all the digits of the number together. If the sum is divisible by 3, then the original number is also divisible by 3. 

If a number meets both of these conditions, then it is divisible by 6. Let’s understand how divisibility by 6 for large numbers works with an example.  

Consider the number 826548.

• First, we check the last digit. Since it ends in 8, which is an even number, it passes the first test and is divisible by 2. Now, add all the digits: 8 + 2 + 6 + 5 + 4 + 8 = 33. 

• Since 33 is divisible by 3, it also passes the second test. 

Because the number is divisible by both 2 and 3, we can conclude that 826548 is divisible by 6.

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Divisibility Rule of 6 and 7

The divisibility rule of 6 and the rule of 7 are different in how they work.

The divisibility rule of 6 is simple. A number is divisible by 6 if it meets two conditions. 

• First, the number must be even. That means the last digit must be 0, 2, 4, 6, or 8. 

• Second, the sum of all the digits in the number must be divisible by 3. If both of these conditions are true, then the number is divisible by 6.

The divisibility rule of 7 is a bit more complicated. To check if a number is divisible by 7, follow this method:

1. Take the last digit of the number.

2. Double that last digit.

3. Subtract the result from the remaining part of the number (ignore the last digit).

4. If the result is divisible by 7, then the original number is divisible by 7.

Let’s take an example using the number 203.

Step 1: The last digit is 3.

Step 2: Double it. 3 × 2 = 6.

Step 3: Subtract 6 from the remaining number, which is 20.

20 − 6 = 14.

Step 4: 14 is divisible by 7, so 203 is divisible by 7.

Divisibility Rule of 6 and 9

Now let us compare the divisibility rule of 6 and the rule of 9.

As we already know, the rule for 6 requires the number to be divisible by both 2 and 3. This means the number must be even, and the sum of its digits must be divisible by 3.

The divisibility rule of 9 is simpler in one way. 

To check if a number is divisible by 9, you only need to do one thing. Add all the digits of the number together. If the total is divisible by 9, then the original number is divisible by 9. It does not matter whether the number is even or odd.

Let’s look at an example using the number 450.

To test if 450 is divisible by 6:

• The last digit is 0, which is even. So it passes the rule for divisibility by 2.

• Add the digits: 4 + 5 + 0 = 9. Since 9 is divisible by 3, the number also passes the divisibility test for 3.

Because 450 is divisible by both 2 and 3, it is divisible by 6.

Now let’s test 450 for divisibility by 9:

• Add the digits again: 4 + 5 + 0 = 9.

• 9 is divisible by 9, so the number 450 is also divisible by 9.

This shows that the number 450 is divisible by both 6 and 9, but the rules used to check them are different.

The divisibility rule of 6 checks two conditions, while the rule of 9 checks only one. Each rule is useful in different situations. The rule of 6 helps you quickly test for numbers that are part of the 6 times table. The rule of 9 is useful when dealing with problems that involve patterns in digit sums.

Read More: Division of Fractions

Divisibility Rule of 6 With Examples

Let’s now understand the divisibility rule of 6 with examples that show how to test both small and large numbers step by step.

Example 1: Is 66 divisible by 6?

Solution:

Step 1: Is it divisible by 2? Yes. It ends in 6, which is even.

Step 2: Add the digits: 6 + 6 = 12. Twelve is divisible by 3.

Since both conditions are true, 66 is divisible by 6.

Example 2: Is 49 divisible by 6?

Solution:

Step 1: It ends in 9, which is odd. So it is not divisible by 2.

Step 2: Add the digits: 4 + 9 = 13. Thirteen is not divisible by 3.

It fails both conditions, so 49 is not divisible by 6.

Example 3: Is 24576 divisible by 6?

Solution:

Step 1: The number ends in 6, which is even. So it is divisible by 2.

Step 2: Add the digits: 2 + 4 + 5 + 7 + 6 = 24. Twenty-four is divisible by 3.

Since it is divisible by both 2 and 3, 24576 is divisible by 6.

Example 4: Is 81325 divisible by 6?

Solution:

Step 1: The number ends in 5, which is odd. So it is not divisible by 2.

Step 2: Add the digits: 8 + 1 + 3 + 2 + 5 = 19. Nineteen is not divisible by 3.

It does not pass either test, so 81325 is not divisible by 6.

Read More: Divisibility Rule of 11

Divisibility Rule of 6 Practice Questions

Here are some numbers. Try checking them using the divisibility rule of 6:

1. 264

2. 80

3. 433788

4. 56760

5. 7890

Answers

• 2 + 6 + 4 = 12 - Even and divisible by 3 - Yes

• 8 + 0 = 8 - Even, but not divisible by 3 - No

• 4 + 3 + 3 + 7 + 8 + 8 = 33 - Even and divisible by 3 - Yes

• Ends in 0 and digits add to 24 - Yes

• Ends in 0 and digits add to 24 - Yes

Keep practicing with more divisibility rules of 6 examples, and you will master it in no time.

Also read: Least Common Multiple

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