# Divisibility Tests

Hello, and welcome to this video about divisibility tests! Today we’ll learn how to figure out if a number is divisible by 2,3,4,5,6,8,9, and 10.

Before we get started, let’s review a few things. When one number is **divisible** by another, it means that the **quotient** is a **whole number**. For instance, 8 is divisible by 4 because \(8\div4=2\). On the other hand, 8 is *not* divisible by 3 because \(8\div3=2.\overline{66}\).

**Divisibility tests** are useful ways to determine if an integer is divisible by a divisor without doing a ton of work. These tests involve examining the digits and looking for patterns.

Let’s look at some divisibility tests for some of the most common divisors.

If a number is divisible by 2, then its last digit is an even number, ending in 0, 2, 4, 6, or 8.

For example, 546 is divisible by 2 because it ends in an even number, 6. 545 is *not* divisible by 2 because it ends in 5, which is not an even number.

If a number is divisible by 3, then the sum of its digits is divisible by 3.

Let’s look at the number 1,635. First, add the digits together: \(1+6+3+5=15\). Next, divide by 3: \(15\div3=5\). Since the sum of the digits, 15, is divisible by 3, 1,635 is also divisible by 3.

Now let’s consider the number 452. To see if 452 is divisible by 3, find the sum of the digits and then divide by 3. \(4+5+2=11\), and \(11\div3=3.\overline{6}\). That means 452 is not divisible by 3.

If a number is divisible by 4, then its last two digits are divisible by 4.

Let’s consider the number 9,316. First, identify the last two digits, 16. Since \(16\div4=4\), 9,316 is divisible by 4.

Now let’s look at the number 45,118. The last two digits of this integer are 18. Since \(18\div4=4.5\), 45,118 is not divisible by 4.

If a number is divisible by 5, then it must end in a 0 or a 5.

For instance, 640 is divisible by 5 because its last digit is 0. On the other hand, 932 is *not* divisible by 5 because its last digit is neither 0 nor 5.

A number is divisible by 6 if it passes the divisibility tests for 2 and 3 that we talked about earlier. In other words, the number must be even and the sum of its digits must be divisible by 3.

Let’s look at the number 108. We know it’s an even number because its last digit, 8, is even. From here, add the digits together to see if the sum is divisible by 3. \(1+0+8=9\), and \(9\div3=3\). Therefore, 108 is divisible by 6.

Let’s try one more. Consider the number 446. We know it’s an even number because its last digit, 6, is even. From here, add the digits together to see if the sum is divisible by 3. \(4+4+6=14\), and \(14\div3=4.\overline{6}\). Therefore, 446 is *not* divisible by 6.

A number is divisible by 8 if its last three digits are divisible by 8.

Let’s consider the number 7,184. Divide the last 3 digits, 184, by 8: \(184\div8=23\), so 7,184 is divisible by 8.

Now let’s look at the number 65,447. Divide the last 3 digits, 447, by 8: \(447\div8=55.875\), so 65,447 is *not* divisible by 8.

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

Let’s take a look at the number 3,627. First, add the digits together: \(3+6+2+7=18\). Then, divide 18 by 9. \(18\div9=2\), so 3,627 is divisible by 9.

What about 2,157? Is this number divisible by 9? Let’s start by adding the digits together: \(2+1+5+7=15\), and 15 is not divisible by 9. Therefore, 2,157 is not divisible by 9.

A number is divisible by 10 if the number ends in 0.

For example, 890 is divisible by 10 because it ends in 0. 891 is not divisible by 10 because it does not end in 0.

## Review

Now that you know the most common divisibility tests, it’s your turn. Consider the following problems:

*Is 405 divisible by 6?**Is 18,524 divisible by 4?**Katy says 105 is divisible by 3, but Jamal says it’s not. Who is correct?*

Pause the video here and try these problems yourself. When you’re done, resume the video, and we’ll go over everything together.

*Is 405 divisible by 6?*

To figure out if 405 is divisible by 6, it must be even and the sum of its digits must be divisible by 3. Since 405 is not an even number, it’s not divisible by 6.*Is 18,524 divisible by 4?*

If 18,524 is divisible by 4, then its last two digits must be divisible by 4. First, identify the last two digits, 24. Since \(24\div4=6\), 18,524 is divisible by 4.*Katy says 105 is divisible by 3, but Jamal says it’s not. Who is correct?*

To figure out if 105 is divisible by 3, first add together the digits: \(1+0+5=6\). Then, determine if the sum is divisible by 3. Since \(6\div3=2\), Katy is correct.

Great job!

I hope this video about divisibility tests was helpful. Thanks for watching, and happy studying!

## Divisibility Test Practice Questions

**Question #1:**

Which of the following numbers is 23,128 divisible by? Why?

3 because the sum of the digits of 23,128 is divisible by 3.

8 because the last three digits of 23,128 are divisible by 8.

9 because the sum of the digits of 23,128 is divisible by 9.

6 because 23,128 is an even number and the sum of its digits is divisible by 3.

**Answer:**

A number is divisible by 8 if its last three digits are divisible by 8. The last three digits of 23,128 are 128. Since \(128\div8=16\), the value 23,128 is divisible by 8.

**Question #2:**

Which of the following numbers is 7,482 divisible by? Why?

9 because the sum of the digits of 7,482 is divisible by 9.

5 because 7,482 does not end in a 0 or a 5.

6 because 7,482 is an even number and the sum of its digits is divisible by 3.

4 because the last two digits in 7,482 are divisible by 4.

**Answer:**

A number is divisible by 6 if it passes the divisibility tests for 2 and 3.

- If a number is divisible by 2, then its last digit is an even number, 0, 2, 4, 6, or 8. Since 7,482 ends in a 2, it is divisible by 2.
- A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 7,482 is:
\(7+4+8+2=21\)

Since \(21\div3=7\), the sum of the digits is divisible by 3, so 7,482 is also divisible by 3.

Since 7,482 passes the divisibility tests for 2 and 3, it is also divisible by 6.

**Question #3:**

Which of the following statements is true? Why?

14,259 is divisible by 9 because the sum of its digits is divisible by 9.

25,493 is divisible by 8 because the last three digits are divisible by 8.

16,075 is divisible by 5 because it ends in a 0 or a 5.

37,732 is divisible by 6 because it is an even number, and the sum of its digits is divisible by 3.

**Answer:**

A number divisible by 5 must end in a 0 or a 5. Since 16,075 ends in a 5, it is divisible by 5.

**Question #4:**

You have been saving quarters ($0.25 each) in a large glass bottle to buy a new bicycle. You empty the bottle to count the number of quarters you have saved. To make the task easier, you divide the quarters into groups of four to see how many whole dollars you have saved. Which of the following could be the number of quarters you have saved if you have saved a whole number of dollars?

492

525

394

446

**Answer:**

There are four quarters in one dollar. If you have saved a whole number of dollars, then the number of quarters you have saved must be divisible by 4.

A number is divisible by 4 if its last two digits are divisible by 4. The last two digits in 492 are 92. Since \(92\div4=23\), the value 492 is divisible by 4. Of the choices provided, you have saved a whole number of dollars when you have saved 492 quarters. (You have actually saved $123 since \(492\div4=123\).)

**Question #5:**

A factory fills bags of dog food and ships the bags in boxes to various stores each day. Each box contains 6 bags of dog food. Given that no partially filled boxes of dog food are shipped, which of the following could be the total number of bags of dog food that are shipped each day?

2,356

1,518

1,245

2,349

**Answer:**

Since no partially filled boxes of dog food are shipped each box contains 6 bags of dog food, the total number of bags of dog food boxed each day must be divisible by 6.

A number is divisible by 6 if it passes the divisibility tests for 2 and 3.

- If a number is divisible by 2, then its last digit is an even number, 0, 2, 4, 6, or 8. Since 1,518 ends in an 8, it is divisible by 2.
- A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 1,518 is:
\(1+5+1+8=15\)

Since \(15\div3=5\), the sum of the digits is divisible by 3, so 1,518 is also divisible by 3.

Since 1,518 passes the divisibility tests for 2 and 3, it is also divisible by 6. So, of the answer choices provided, the factory could ship a total of 1,518 bags of dog food each day. (The factory ships 253 boxes of dog food each day since \(1{,}518\div6=253\).)