Lesson Objectives
• Demonstrate an understanding of the division operation
• Demonstrate an understanding of remainders
• Learn the divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10, and 12
• Learn the divisibility rules for 7 and 11

## What are the Divisibility Rules

A number is said to be divisible by another if the result of the division has no remainder. As an example, we can say that 15 is divisible by 3 since 15 ÷ 3 is 5 with no remainder. In this lesson, we will learn some of the divisibility rules. These rules will help us to quickly determine if one number is divisible by another. Let's first take a look at the divisibility rules for 2, 3, 4, 5, 6, 8, 10, and 12. We will leave the divisibility rules for 7 and 11 for the end, as these require additional explanation.

### Divisibility by 2

• A number is divisible by 2 if the number is even. This means the final digit of the number ends with a (0, 2, 4, 6, or 8)

### Divisibility by 3

• A number is divisible by 3 if the sum of the number's digits is divisible by 3

### Divisibility by 4

• A number is divisible by 4 if the final two digits of the number form a number that is divisible by 4

### Divisibility by 5

• A number is divisible by 5 if the final digit is a 0 or a 5

### Divisibility by 6

• A number is divisible by 6 if the number is divisible by both 2 and 3

### Divisibility by 8

• A number is divisible by 8 if the final three digits of the number form a number that is divisible by 8

### Divisibility by 9

• A number is divisible by 9 if the sum of the number's digits is divisible by 9

### Divisibility by 10

• A number is divisible by 10 if the final digit is a 0

### Divisibility by 12

• A number is divisible by 12 if the number is divisible by both 3 and 4
Let's take a look at a few examples.
Example 1: Determine whether each number is divisible by: 2, 3, 4, 5, or 6.
639
• 2 - No, the final digit is a 9
• 3 - Yes, the sum of the digits results in: 6 + 3 + 9 = 18, which is divisible by 3. 18 ÷ 3 = 6
• 4 - No, the final two digits of the number form the number 39: 39 is not divisible by 4, 39 ÷ 4 = 9 R3
• 5 - No, the final digit of the number is not a 0 or 5
• 6 - No, the number is not divisible by both 2 and 3
120
• 2 - Yes, the final digit is a 0
• 3 - Yes, the sum of the digits results in: 1 + 2 + 0 = 3, which is divisible by 3. 3 ÷ 3 = 1
• 4 - Yes, the final two digits of the number form the number 20: 20 is divisible by 4, 20 ÷ 4 = 5
• 5 - Yes, the final digit of the number is 0
• 6 - Yes, the number is divisible by both 2 and 3
330
• 2 - Yes, the final digit is a 0
• 3 - Yes, the sum of the digits results in: 3 + 3 + 0 = 6, which is divisible by 3. 6 ÷ 3 = 2
• 4 - No, the final two digits of the number form the number 30: 30 is not divisible by 4, 30 ÷ 4 = 7 R2
• 5 - Yes, the final digit of the number is 0
• 6 - Yes, the number is divisible by both 2 and 3
Example 2: Determine whether each number is divisible by: 8, 9, 10, or 12.
1352
• 8 - Yes, the final three digits of the number form the number 352, which is divisible by 8. 352 ÷ 8 = 44
• 9 - No, the sum of the digits results in: 1 + 3 + 5 + 2 = 11, which is not divisible by 9. 11 ÷ 9 = 1 R2
• 10 - No, the final digit of the number is not a 0
• 12 - No, the number is not divisible by 3
9360
• 8 - Yes, the final three digits of the number form the number 360, which is divisible by 8. 360 ÷ 8 = 45
• 9 - Yes, the sum of the digits results in: 9 + 3 + 6 + 0 = 18, which is divisible by 9. 18 ÷ 9 = 2
• 10 - Yes, the final digit of the number is a 0
• 12 - Yes, the number is divisible by both 3 and 4
57,684
• 8 - No, the final three digits of the number form the number 684, which is not divisible by 8. 684 ÷ 8 = 85 R4
• 9 - No, the sum of the digits results in: 5 + 7 + 6 + 8 + 4 = 30, which is not divisible by 9. 30 ÷ 9 = 3 R3
• 10 - No, the final digit of the number is not a 0
• 12 - Yes, the number is divisible by both 3 and 4
When we think about the rules for 7 and 11, they can be a bit more complex. Let’s think about divisibility by 7 first, and then we will discuss the rule for divisibility by 11.

### Divisibility by 7

• We start by removing the final digit from the number, double the number and subtract it away from the shortened original number
• If the number formed is a 0 or a number that is divisible by 7, the original number is divisible by 7. We can repeat the process as needed
Example 3: Determine if each number is divisible by 7.
46,683
• We start by removing the final digit from the number 46,683, double the number and subtract it away from the shortened original number
• Remove 3 and double the number (multiply by 2) 3 x 2 = 6
• Subtract 6 away from 4668, 4668 - 6 = 4662
• If the number formed is a 0 or a number that is divisible by 7, the original number is divisible by 7. We can repeat the process as needed
• Since the number 4662 is a bit large, let's repeat the process
• We start by removing the final digit from the number 4662, double the number and subtract it away from the shortened original number
• Remove 2 and double the number (multiply by 2) 2 x 2 = 4
• Subtract 4 away from 466, 466 - 4 = 462
• The number 462 is still a little big, let's repeat the process
• We start by removing the final digit from the number 462, double the number and subtract it away from the shortened original number
• Remove 2 and double the number (multiply by 2) 2 x 2 = 4
• Subtract 4 away from 46, 46 - 4 = 42
• We should know that 42 ÷ 7 = 6. Therefore 46,683 is divisible by 7
57,855
• We start by removing the final digit from the number 57,855, double the number and subtract it away from the shortened original number
• Remove 5 and double the number (multiply by 2) 5 x 2 = 10
• Subtract 10 away from 5785, 5785 - 10 = 5775
• If the number formed is a 0 or a number that is divisible by 7, the original number is divisible by 7. We can repeat the process as needed
• Since the number 5775 is a bit large, let's repeat the process
• We start by removing the final digit from the number 5775, double the number and subtract it away from the shortened original number
• Remove 5 and double the number (multiply by 2) 5 x 2 = 10
• Subtract 10 away from 577, 577 - 10 = 567
• The number 567 is still a little big, let's repeat the process
• We start by removing the final digit from the number 567, double the number and subtract it away from the shortened original number
• Remove 7 and double the number (multiply by 2) 7 x 2 = 14
• Subtract 14 away from 56, 56 - 14 = 42
• We should know that 42 ÷ 7 = 6. Therefore 57,855 is divisible by 7

### Divisibility by 11

• Form the sum of the digits in the odd places and subtract away the sum of the digits in the even places
• The odd places refer to the 1st digit, 3rd digit, 5th digit, and so on
• The even places refer to the 2nd digit, 4th digit, 6th digit, and so on
• Note, we can count our digits starting from right to left or left to right. This can result in a sign difference, which is not important.
• If the result is a 0 or a number that is divisible by 11, then the original number is divisible by 11
Example 4: Determine if each number is divisible by 11.
206,910
• Form the sum of the digits in the odd places and subtract away the sum of the digits in the even places
• The odd places refer to the 1st digit, 3rd digit, 5th digit, and so on
• Odd digits sum to: 2 + 6 + 1 = 9
• The even places refer to the 2nd digit, 4th digit, 6th digit, and so on
• Even digits sum to: 0 + 9 + 0 = 9
• Subtract 9 - 9 = 0
• If the result is a 0 or a number that is divisible by 11, then the original number is divisible by 11
• The result is 0, so 206,910 is divisible by 11
786,240
• Form the sum of the digits in the odd places and subtract away the sum of the digits in the even places
• The odd places refer to the 1st digit, 3rd digit, 5th digit, and so on
• Odd digits sum to: 7 + 6 + 4 = 17
• The even places refer to the 2nd digit, 4th digit, 6th digit, and so on
• Even digits sum to: 8 + 2 + 0 = 10
• Subtract 17 - 10 = 7
• If the result is a 0 or a number that is divisible by 11, then the original number is divisible by 11
• The result is 7, 7 is not 0 or divisible by 11, so 786,240 is not divisible by 11

#### Skills Check:

Example #1

Determine if the given number is divisible by 3.

86,992

Please choose the best answer.

A
Yes
B
No

Example #2

Determine if the given number is divisible by 9.

78,678

Please choose the best answer.

A
Yes
B
No

Example #3

Determine if the given number is divisible by 11.

32,142

Please choose the best answer.

A
Yes
B
No

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