Remember the sixth grade social from the Find Factor Pairs of Given Numbers Concept?

Remember that there were two groups of students attending the social, one with 44 students in it and one with 48 in it. These numbers are small for divisibility rules, but what if the class sizes were larger? Imagine that each class had 144 students and the other had 148 students.

How could the divisibility rules help with this dilemma?

**In this Concept, you will learn about divisibility rules so that you can apply it to the dilemma of the sixth grade social.**

### Guidance

When we have a larger number that we are factoring, we may need to use ** divisibility rules** to help us find the factors of that number.

**What are divisibility rules?**

Divisibility rules help determine if a number is divisible by let’s say 2 or 3 or 4. This can help us to identify the factors of a number. Here is a chart that shows all of the basic divisibility rules.

Now some of these rules are going to be more useful than others, but you can use this chart to help you.

What numbers is 1346 divisible by?

To solve this, we can go through each rule and see if it applies.

- The last digit is even-this number is divisible by 2.
- The sum of all the digits is 14-this number is not divisible by 3.
- The last two digits are not divisible by 4-this number is not divisible by 4.
- The last digit is not zero or five-this number is not divisible by 5.
- 1346 - 12 = 1334-this number is not divisible by 7.
- The last three numbers are not divisible by 8.
- The sum of the digits is 14-this number is not divisible by 9
- The number does not end in zero-this number is not divisible by 10
- The number is not divisible by 3 and 4

**Our answer is that this number is divisible by 2.**

**Whew! That is a lot of work! You won’t usually have to go through each rule of divisibility, but it is important that you know and understand them just in case.**

Now it's time for you to practice. Answer the following questions.

#### Example A

Is 3450 divisible by 10?

**Solution: Yes, because it ends in zero it is divisible by ten.**

#### Example B

**Is 1298 divisible by 3?**

**Solution: No, it is not divisible by 3.**

#### Example C

**Is 3678 divisible by 2?**

**Solution: Yes, because it ends in an even number it is divisible by 2.**

Now let's think about the sixth grade social. Do you know how to work with those large numbers?

One class has 144 students and one has 148 students. Let's apply the rules of divisibility to find the factors so that the groups can be easily organized.

First, we can start with 144.

144 ends in an even number, so it is divisible by 2. The last two digits are divisible by 4, so it is divisible by 4.

148 ends in an even number, so it is divisible by 2. The last two digits are divisible by 4, so it is divisible by 4.

It makes sense to divide these two classes into four groups each.

If we then divide both 144 and 148 by 4, we will have the number of students in each group.

144 will have 36 students in each group.

148 will have 37 students in each group.

**This is the solution to the problem.**

### Vocabulary

- Factors
- numbers multiplied together to equal a product.

- Divisibility Rules
- a list of rules which help you to determine if a number is evenly divisible by another number.

### Guided Practice

Here is one for you to try on your own.

Is 918 divisible by 9? Why or why not.

**Answer**

To figure this out, we can use the divisibility rules from the Concept.

For a number to be divisible by 9, we can complete a simple test.

We add up the digits and see if the sum of the digits is divisible by 9. If so, then the entire number is divisible by 9.

\begin{align*}9 + 1 + 8 = 18\end{align*}

**18 is divisible by 9, therefore 918 is also divisible by 9.**

### Video Review

James Sousa Divisibility Rules

- http://www.mathplayground.com/howto_divisibility.html – This is a video that explains divisibility rules.

### Practice

Directions: Answer each question using the rules of divisibility. Explain your reasoning.

1. Is 18 divisible by 3?

2. Is 22 divisible by 2?

3. Is 44 divisible by 6?

4. Is 112 divisible by 2 and 3?

5. Is 27 divisible by 9 and 3?

6. Is 219 divisible by 9?

7. Is 612 divisible by 2 and 3?

8. Is 884 divisible by 4?

9. Is 240 divisible by 5?

10. Is 782 divisible by 7?

11. Is 212 divisible by 4 and 6?

12. Is 456 divisible by 6 and 3?

13. Is 1848 divisible by 8 and 4?

14. Is 246 divisible by 2?

15. Is 393 divisible by 3?

16. Is 7450 divisible by 10?