# Divisibility Rules for 3, 6, and 9

🏆Practice signs of division in 3, 6 and 9

### Divisibility criteria for$3$,$6$ and$9$

#### Divisibility criteria for$3$

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

#### Divisibility criteria for$6$

A number is divisible by $6$ if it is even and also a multiple of $3$.

#### Divisibility criteria for$9$

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

## Test yourself on signs of division in 3, 6 and 9!

Determine if the following number is divisible by 3:

$$352$$

## Divisibility Rules for 3, 6, and 9

Wow! What a pleasant and entertaining topic! In this article, we will teach you how to identify if a number is divisible by $3$, $6$ and $9$, in a matter of seconds!
Shall we start?

## Divisibility Rules for 3

A number is divisible by $3$ if the sum of its digits is a multiple of $3$.
If the sum of the digits of the number is not a multiple of $3$, neither will the original number be.

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### Example 1

The number $714$
How will we know if it is divisible by $3$? In a very simple way, we will calculate the sum of its digits:
$7+1+4=12$

We already know that $12$ is divisible by $3$, therefore, $714$ is as well.

Note: we recommend adding the digits one more step to avoid errors.
That is, if after adding the digits the result is $12$, we can add the new digits obtained again.
$1+2=3$

This sum will give us a smaller number and, in this way, we can be sure whether it is a multiple of $3$ or not.
In the same way, we will know that, if the number obtained in the result is a multiple of $3$, the original one is as well.

### example 2

Is the number $465$ divisible by $3$?
Solution: Let's check the sum of its digits:
$4+6+5=15$

$15$ –> The result is, indeed, a number divisible by $3$, therefore, the original $465$ is as well.
Note: We could have continued and added the digits to arrive at a smaller number.
$1+5=6$
$6$ is divisible by $3$. Therefore, $465$ is also divisible by $3$.

#### Example 3

Is the number $2547$ divisible by $3$?

Solution:
$2+5+4+7=18$
$1+8=9$
$9$ is divisible by $3$, therefore, $2547$ is divisible by $3$.

Do you know what the answer is?

### Example 4

Is the number $8125$ divisible by $3$?

Solution:
$8+1+2+5=16$
$1+6=7$
$7$ is not divisible by $3$, therefore, $2547$ is not divisible by $3$.

## Divisibility Rules for 6

A number is divisible by $6$ if it is even and also a multiple of $3$.
In fact, we must check the $2$ conditions:

• Let's ask if the number is even, for that we can observe the last digit and, if it is even, the whole number is.
• Let's ask if the number is a multiple of $3$. According to what we have learned, a number is divisible by $3$ if the sum of its digits is a multiple of $3$.

If both conditions are met, the number is divisible by $6$.

### Example 1

Is the number $714$ divisible by $6$?

Solution:
Let's see if the number is even.
Yes, the number is even. The units digit is $6$ and $6$ is an even number.
Let's continue with the second condition -> Is the number divisible by $3$?

### Example 2

Let's calculate the sum of its digits:
$7+1+4=12$
$1+2=3$
$3$ is divisible by $3$, therefore, $714$ is also divisible by $3$.
Both conditions are met, so $714$ is divisible by $6$.

Is the number $9081$ divisible by $6$?

Solution:
Let's see if the number is even:
The units digit is $1$, $1$ is odd, therefore, the number is not divisible by $6$.
Even if only one of the conditions is not met, that is enough to determine that the number is not divisible by $6$.

Do you think you will be able to solve it?

## Divisibility Rules for 9

A number is divisible by $9$ if the sum of its digits is a multiple of $9$.
If the sum of the digits of the number is not a multiple of $9$, then the original number will not be either.
Note: After adding the digits once and obtaining some number as a result, it is advisable to also add the digits of this last number to arrive at a smaller number that makes it easier to check if it is a multiple of $9$.

### For example

The number $864$

Solution :
Let's add its digits $8+6+4=18$
$18$ is divisible by $9$ and at this stage, we can determine that $864$ is divisible by $9$.
If you still doubt that $18$ is divisible by $9$, you can add the digits of the result obtained again:
$1+8=9$
$9$ is divisible by $9$, therefore, $864$ is divisible by $9$.

### Example 2

Is the number $8134$ divisible by $9$?

Solution :
$8+1+3+4=16$
$1+6=7$
$7$ is not divisible by $9$, therefore, $8134$ is divisible by $9$.

### Example 3

Is the number $9945$ divisible by $9$?

Solution:
$9+9+4+5=27$
$2+7=9$
$9$ is divisible by $9$, therefore, $9945$ is divisible by $9$.

## Examples and exercises with solutions for divisibility rules for 3, 6, and 9

### Exercise #1

Determine if the following number is divisible by 3:

$352$

No

### Exercise #2

Determine if the following number is divisible by 3:

$673$

No

### Exercise #3

Will a number divisible by 6 necessarily be divisible by 2?

Yes

### Exercise #4

Determine if the following number is divisible by 3:

$132$

Yes

### Exercise #5

Determine if the following number is divisible by 3:

$564$