# Divisibility Rules - 3, 6, 9

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Eugene L.

## Basics on the topicDivisibility Rules - 3, 6, 9

Divisibility rules help us determine if a number will divide into another number without actually having to divide. Any number which satisfies a divisibility rule for a given number is divisible by that given number. Otherwise, it is not divisible by that given number. There is a divisibility rule for every number. Learn the divisibility rules for 3, 6, and 9 by playing Creature Capture. Common Core Reference: CCSS.MATH.CONTENT.6.NS.B.4

### TranscriptDivisibility Rules - 3, 6, 9

There’s a hot new game that’s got everyone playing! It’s called: Creature Capture. Each player’s objective is to capture and then train the creatures they find. Players receive points for victories and can use those points in order to level up their creatures. Meet Olivia. She’s been playing Creature Capture for a few weeks now.

She’s just finished a battle with one of her friends and has some experience points to spend. The creatures in the game can be leveled up in three categories: Health, Attack and Defense. There’s just one catch: Players can only add points to the different categories using the Divisibility Rules for 3, 6 and 9. Players can increase the Health of their characters if the total number of points they have is divisible by 3.They can increase their Attack if their point total is divisible by 6. And finally, if their total is divisible by 9, players can increase their character’s defense.

Olivia has 348 points to spend, but which of her character’s abilities can Olivia increase?Let’s help Olivia upgrade her character. Is 348 divisible by 3? 6? 9? Is there a quick way to find out without having to use long division each time? Yes!

### The Divisibility Rule for 3

There are so-called "divisibility rules" that help you determining whether a certain number is divisible by a number 1-10. The divisibility rules for 3 and 9 are quite similar. First, find the sum of all the digits in the number. For 348, you have to add 3 + 4 + 8. Doing so gives us 15.

If the resulting sum is divisible by 3, then the whole number is also divisible by 3. In this case, 3 goes into 15 exactly 5 times, so we can say that 15 is divisible by 3. This also means that 348 is divisible by 3!

### The Divisibility Rule for 9

Olivia can upgrade her creature’s Health. But to figure out all her options, she checks if she can upgrade her creature's defense instead. Again, find the sum of all the digits in the number. As we already know, 3 + 4 + 8 is equal to 15. Since 15 divided by 9 gives you 1 and a remainder of 6, you know it isn't evenly divisible by 9. So we know that 348 is not divisible by 9. So Olivia cannot upgrade her creature's defense. Olivia is curious if she can level up her character's Attack.

### The Divisibility Rule for 9

If Olivia wants her character to pack a more powerful punch, 348 has to be divisible by 6. But, how can you tell if a number is divisible by 6? It's easy! If a number is divisible by 2 and 3, it is also divisible by 6. First, check to see if the number is even. If it is, you can be sure that it's also divisible by 2. Since 348 ends in an 8, 348 is an even number! Next, check to see if the number is divisible by 3. You know already that 348 is divisible by 3. So, because the answer to both of these questions is yes, 348 is divisible by 6. Olivia can either level up Attack, or health.

### Summary of the Divisibility Rules

Let's summarize the divisibility rules:

• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 9 if the sum of its digits is divisible by 9.
• And a number is divisible by 6 if it is divisible by 2 (even number) and by 3. Back to the game.

Olivia decides to increase her character’s Attack. She just got challenged by a gamer who goes by the tag -=13urntK00k!ez=-. Let’s see how well Olivia does with her improved character!
Oh no, he attacks first. What an impressive attack!! Maybe Olivia should have upgraded her character's Health or Defense instead. Who is this gifted gamer? Granny?

## Divisibility Rules - 3, 6, 9 exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Divisibility Rules - 3, 6, 9.
• ### Recall the divisibility rules for 3, 6, and 9.

Hints

The first step in the divisibility rules for $3$ and $9$ is to sum all of the digits in the number.

A number that is divisible by $2$ and $3$ is also divisible by $6$.

Solution

Divisibility Rule for 3

The first step in checking if a number is divisible by three is to sum all the digits. If the resulting sum is divisible by $3$, then the original number is also divisible by $3$. Here's an example with the number $348$:

$3 + 4 + 8 = 15$.

$15$ is divisible by $3$, so $348$ must also be divisible by $3$.

$~$

Divisibility Rule for 6

If a number is divisible by $2$ and $3$, then it is also divisible by $6$. To check if a number is divisible by two, look at its last digit. If the last digit is $0$, $2$, $4$, $6$ or $8$, then the number is divisible by $2$. To check if a number is divisible by three, follow the instructions above.

For example, the number $348$ is divisible by $2$ because the last digit is $8$. We already found that it is also divisible by $3$, so it must be divisible by $6$ as well.

$~$

Divisibility Rule for 9

Just like the divisibility rule for $3$, the first step in checking if a number is divisible by $9$ is to sum all the digits. If the resulting sum is divisible by $9$, then the original number is divisible by $9$. Using the number $348$ as an example:

$3 + 4 + 8 = 15$.

$15$ is not divisible by $9$, so neither is $348$.

• ### Decide if numbers are divisible by 2, 3, 6, or 9.

Hints

Try using the divisibility rules for $3$, $6$, and $9$ to check if a number is divisible by those numbers.

$9$ and $6$ are both divisible by $3$.

Therefore, if a number is not divisible by $3$, then it cannot be divisible by $6$ or $9$.

It'll be fastest to start by checking if each number is divisible by $3$. If it isn't divisible by $3$, then you already know that it's not divisible by $3$, $6$, and $9$.

Solution

Divisibility Rule for 3

The first step in checking if a number is divisible by three is to sum all the digits. If the resulting sum is divisible by $3$, then the original number is also divisible by $3$. For example, with the number $128$:

$1 + 2 + 8 = 11$

$11$ is not divisible by $3$, so $128$ is not divisible by $3$ either.

$~$

Divisibility Rule for 6

If a number is divisible by $2$ and $3$, then it is also divisible by $6$. To check if a number is divisible by two, look at its last digit. If the last digit is $0$, $2$, $4$, $6$, or $8$, then the number is divisible by $2$. $128$ is divisible by $2$ because the last digit is $8$. But we already found that is is not divisible by $3$. So it cannot be divisible by $6$ either.

$~$

Divisibility Rule for 9

Just like the divisibility rule for $3$, the first step in checking if a number is divisible by $9$ is to sum all the digits. If the resulting sum is divisible by $9$ then the original number is divisible by $9$. So using the number $128$ as an example:

$1 + 2 + 8 = 11$

$11$ is not divisible by $9$, so neither is $348$.

• ### Explain how Olivia can level up if she has 1356 points.

Hints

Look at this example, using the number $1234$ and the sum of its digits:

$\quad~~~1+2+3+4=10$

This sum isn't divisible by $3$ and thus $1234$ is neither divisible by $3$ nor by $6$ nor by $9$.

$1356$ is divisible by $2$ of the following numbers: $2$, $3$, $6$, and $9$.

Solution

For checking divisibility, follow these steps:

1. Check the divisibility rule for $3$: The sum of the digits $1+3+5+6=15$ must be divisible by $3$. It sure is: $15\div 3=5$. Thus $1356$ is divisible by $3$ too.
2. Is the given number even or odd? $1356$ ends with $6$, an even number, so $1356$ is also even and thus divisible by $2$. If any number is divisible by $2$ as well as $3$, it's divisible by $6$ too.
3. Again take a look at the sum of the digits: $15$. It's not evenly divisible by $9$. Therefore, we can conclude that $1356$ is not divisible by $9$.
That means Olivia could either improve her Creature's health or attack level.

• ### Calculate the different ways in which Olivia’s grandmother can level up if she has 5634 points.

Hints

The digit sum is the sum of all individual digits of a given number. Let's have a look at an example, $2345$:

$\quad~~~2+3+4+5=14$

A number is divisible by $3$ (or $9$) if the digit sum is divisible by $3$ (or $9$).

Even numbers end in $0$, $2$, $4$, $6$, or $8$.

All even numbers are divisible by $2$.

Solution

You want to check the divisibility of the number $5643$.

1. First, sum the digits: $5+6+4+3=18$. This sum is divisible by $3$ as well as $9$. Thus you can conclude that the number $5643$ is divisible by $3$ and $9$.
2. Next, take a look at the last digit: It's $6$ and thus even. $5643$ is divisible by $2$.
3. Together with the divisibility by $3$ you know that $5643$ is divisible by $6$, too.
Great job by Olivia's Grandma. She could improve all of her Creature's levels: health, attack and defense. She just has to decide which one!

• ### Determine if $348$ is divisible by 3, 6, or 9.

Hints

You can use the divisibility rules for $3$, $6$, and $9$ to complete this task.

The divisibility rules for $3$ and $9$ are similar.

The first step in the divisibility rules for $3$ and $9$ is to sum all of the digits in the number.

Each number which is divisible by $2$ and $3$ is divisible by $6$, too.

Solution

Divisibility Rule for 3

The first step in checking if a number is divisible by three is to sum all the digits. If the resulting sum is divisible by $3$, then the original number is also divisible by $3$. For example, with the number $348$:

$3 + 4 + 8 = 15$

$15$ is divisible by $3$, so $348$ must also be divisible by $3$.

$~$

Divisibility Rule for 6

If a number is divisible by $2$ and $3$, then it is also divisible by $6$. To check if a number is divisible by two, look at its last digit. If the last digit is $0$, $2$, $4$, $6$, or $8$, then the number is divisible by $2$. To check if a number is divisible by three, follow the instructions above.

So, the number $348$ is divisible by $2$ because the last digit is $8$. We already found that is is also divisible by $3$, so it must be divisible by $6$ as well.

$~$

Divisibility Rule for 9

Just like the divisibility rule for $3$, the first step in checking if a number is divisible by $9$ is to sum all the digits. If the resulting sum is divisible by $9$, then the original number is divisible by $9$. Using the number $348$ as an example:

$3 + 4 + 8 = 15$

$15$ is not divisible by $9$, so neither is $348$.

• ### Figure out the different ways Olivia can level up if she has a certain number of points.

Hints

Each number divisible by $6$ must be an even number.

A number is divisible by $3$ if the digit sum is divisible by $3$.

The rule for divisibility by $9$ is quite similar.

You can assign three numbers to each central element.

Solution

Let's start with divisibility by $9$: for a number to be divisible by 9, the digit sum must also be divisible by $9$.

• $657$ leads to the digit sum $6+5+7=18$, which is divisible by $9$. Because $657$ is odd, it's surely not divisible by $6$.
• $234567$ leads to the digit sum $2+3+4+5+6+7=27$. And $27\div 9=3$ so $234567$ is divisible by $9$. This number is also odd and thus not divisible by $6$.
• $3339$ gives us the digit sum $3+3+3+9=18$. Because this is divisible by $9$ we can conclude, together with the ending $9$, $3339$ is divisible by $9$ but not by $6$.
There are still six numbers left. We determine the digit sum for each:

• $636$ $\rightarrow$ $6+3+6=15$ which is divisible by $3$. The number is even and thus divisible by $6$.
• $123456$ $\rightarrow$ $1+2+3+4+5+6=21$. This is divisible by $3$ too. Again, the number is even, and so we can conclude the divisibility by $6$ once again.
• $3336$ $\rightarrow$ $3+3+3+6=15$ is divisible by $3$. Together with the ending $6$, which indicates that the number is even, this shows us that it is divisible by $6$.
• $746$ $\rightarrow$ $7+4+6=17$. This digit sum isn't divisible by $3$, and thus $746$ is neither divisible by $6$ nor by $9$.
• $234566$ $\rightarrow$ $2+3+4+5+6+6=26$. Using the same process as above, this number is neither divisible by $6$ nor by $9$.
• $3335$ $\rightarrow$ $3+3+3+5=14$. This number also isn't divisible by $6$ or $9$.