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

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

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

Determine if the following number is divisible by 3:

\( 352 \)

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?

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.

Test your knowledge

Question 1

Determine if the following number is divisible by 3:

\( 673 \)

Question 2

Will a number divisible by 6 necessarily be divisible by 3?

Question 3

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

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.

**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 $3$.

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?

Question 1

Determine if the following number is divisible by 3:

\( 132 \)

Question 2

Determine if the following number is divisible by 3:

\( 564 \)

Question 3

Is knowing the multiplication table necessary in order to know which numbers are divisible by 6?

**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 $3$.

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 $6$.

Check your understanding

Question 1

Will a number divisible by 9 necessarily be divisible by 6?

Question 2

Will a number divisible by 3 necessarily be divisible by 6?

Question 3

Will a number divisible by 3 necessarily be divisible by 9?

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 $3$?

**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?

Question 1

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

Question 2

Is the number below divisible by 9?

\( 987 \)

Question 3

Is the number below divisible by 9?

\( 685 \)

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 $9$.

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 $9$.

Test your knowledge

Question 1

Will a number divisible by 9 necessarily be divisible by 3?

Question 2

Is the number below divisible by 9?

\( 189 \)

Question 3

Determine if the following number is divisible by 3:

\( 352 \)

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 $9$.

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 $9$.

Determine if the following number is divisible by 3:

$352$

No

Determine if the following number is divisible by 3:

$673$

No

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

Yes

Determine if the following number is divisible by 3:

$132$

Yes

Determine if the following number is divisible by 3:

$564$

Yes

Do you know what the answer is?

Question 1

Determine if the following number is divisible by 3:

\( 673 \)

Question 2

Will a number divisible by 6 necessarily be divisible by 3?

Question 3

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

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