Divisibility Rules: A Comprehensive Guide

Divisibility rules are mathematical shortcuts that allow us to quickly determine whether a given number is divisible by another number without performing a full division operation.

In this article, we will provide you with easy-to-understand step-by-step instructions for applying these rules to various numbers. Let’s dive into the world of divisibility rules!

Divisibility Rule for 2

To determine if a number is divisible by 2, simply check the last digit of the number. If the last digit is an even number (0, 2, 4, 6, or 8), then the number is divisible by 2.

Example:

Is 314 divisible by 2?

• Step 1: Check the last digit – The last digit is 4.
• Step 2: Since 4 is an even number, 314 is divisible by 2.

Divisibility Rule for 3

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

Example:

Is 153 divisible by 3?

• Step 1: Find the sum of the digits – 1 + 5 + 3 = 9.
• Step 2: Check if the sum (9) is divisible by 3 – Yes, it is.
• Step 3: Therefore, 153 is divisible by 3.

Divisibility Rule for 4

A number is divisible by 4 if the last two digits of the number form a number that is divisible by 4.

Example:

Is 1,256 divisible by 4?

• Step 1: Check the last two digits – 56.
• Step 2: Determine if 56 is divisible by 4 – Yes, it is (56 ÷ 4 = 14).
• Step 3: Therefore, 1,256 is divisible by 4.

Divisibility Rule for 5

A number is divisible by 5 if its last digit is either 0 or 5.

Example:

Is 625 divisible by 5?

• Step 1: Check the last digit – The last digit is 5.
• Step 2: Since the last digit is 5, 625 is divisible by 5.

Divisibility Rule for 6

A number is divisible by 6 if it is divisible by both 2 and 3.

Example:

Is 126 divisible by 6?

• Step 1: Check if the number is divisible by 2 – The last digit is 6, which is an even number. Therefore, it is divisible by 2.
• Step 2: Check if the number is divisible by 3 – The sum of the digits is 1 + 2 + 6 = 9, which is divisible by 3. Therefore, it is divisible by 3.
• Step 3: Since 126 is divisible by both 2 and 3, it is also divisible by 6.

Divisibility Rule for 7

The divisibility rule for 7 is less straightforward than the rules for smaller numbers. To check if a number is divisible by 7, follow these steps:

1. Separate the last digit from the rest of the number.
2. Double the separated digit.
3. Subtract the result from the remaining part of the number.
4. Repeat steps 1-3 until you get a smaller number that you can easily determine if it is divisible by 7 or not.

Example:

Is 679 divisible by 7?

• Step 1: Separate the last digit (9) from the rest of the number (67).
• Step 2: Double the last digit (9) – 9 * 2 = 18.
• Step 3: Subtract the result (18) from the remaining part of the number (67) – 67 – 18 = 49.
• Step 4: Check if the new number (49) is divisible by 7. Since 7 * 7 = 49, the number 679 is divisible by 7.

Keep in mind that this method may require multiple iterations for larger numbers. Nonetheless, it provides an effective way to determine if a number is divisible by 7 without using long division.

Divisibility Rule for 8

The divisibility rule for 8 is useful when trying to determine if a number is divisible by 8 without performing long division. To check if a number is divisible by 8, follow these steps:

1. Look at the last three digits of the number.
2. Check if the number formed by the last three digits is divisible by 8.

If the number formed by the last three digits is divisible by 8, then the entire number is divisible by 8.

Example:

Is 4,408 divisible by 8?

• Step 1: Look at the last three digits – 408.
• Step 2: Check if the number 408 is divisible by 8. Since 8 * 51 = 408, the number 4,408 is divisible by 8.

Divisibility Rule for 9

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

Example:

Is 1,836 divisible by 9?

• Step 1: Find the sum of the digits – 1 + 8 + 3 + 6 = 18.
• Step 2: Check if the sum (18) is divisible by 9 – Yes, it is (18 ÷ 9 = 2).
• Step 3: Therefore, 1,836 is divisible by 9.

Divisibility Rule for 10

A number is divisible by 10 if its last digit is 0.

Example:

Is 520 divisible by 10?

• Step 1: Check the last digit – The last digit is 0.
• Step 2: Since the last digit is 0, 520 is divisible by 10.

Divisibility Rule for 11

A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or a multiple of 11.

Example:

Is 2,948 divisible by 11?

• Step 1: Find the sum of the digits in odd positions – 2 + 4 = 6.
• Step 2: Find the sum of the digits in even positions – 9 + 8 = 17.
• Step 3: Calculate the difference between the sums – 17 – 6 = 11.
• Step 4: Check if the difference (11) is a multiple of 11 or 0 – Yes, it is a multiple of 11.
• Step 5: Therefore, 2,948 is divisible by 11.

Divisibility Rule for 12

A number is divisible by 12 if it is divisible by both 3 and 4.

Example:

Is 336 divisible by 12?

• Step 1: Check if the number is divisible by 3 – The sum of the digits is 3 + 3 + 6 = 12, which is divisible by 3. Therefore, it is divisible by 3.
• Step 2: Check if the number is divisible by 4 – The last two digits are 36, which is divisible by 4 (36 ÷ 4 = 9). Therefore, it is divisible by 4.
• Step 3: Since 336 is divisible by both 3 and 4, it is also divisible by 12.

Divisibility Rule for 13

The divisibility rule for 13 is slightly more complex than the previous rules. To determine if a number is divisible by 13, follow these steps:

1. Separate the last digit from the rest of the number.
2. Multiply the separated digit by 4.
3. Add the result to the remaining part of the number.
4. Repeat steps 1-3 until you get a smaller number that you can easily determine if it is divisible by 13 or not.

Example:

Is 507 divisible by 13?

• Step 1: Separate the last digit (7) from the rest of the number (50).
• Step 2: Multiply the last digit (7) by 4 – 7 * 4 = 28.
• Step 3: Add the result (28) to the remaining part of the number (50) – 50 + 28 = 78.
• Step 4: Check if the new number (78) is divisible by 13. Since 13 * 6 = 78, the number 507 is divisible by 13.

Remember, this method may require a few iterations for larger numbers. However, it is still an effective way to determine if a number is divisible by 13 without resorting to long division.

Divisibility Rules Table

Here is a quick reference table for the divisibility rules we’ve covered in this article:

Divisor	Divisibility Rule
2	A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8).
3	A number is divisible by 3 if the sum of its digits is divisible by 3.
4	A number is divisible by 4 if the last two digits form a number that is divisible by 4.
5	A number is divisible by 5 if its last digit is either 0 or 5.
6	A number is divisible by 6 if it is divisible by both 2 and 3.
7	Separate the last digit, double it, and subtract from the remaining part. Repeat if needed.
8	A number is divisible by 8 if the last three digits form a number that is divisible by 8.
9	A number is divisible by 9 if the sum of its digits is divisible by 9.
10	A number is divisible by 10 if its last digit is 0.
11	The difference between the sum of odd and even position digits is either 0 or a multiple of 11.
12	A number is divisible by 12 if it is divisible by both 3 and 4.
13	Separate the last digit, multiply it by 4, and add to the remaining part. Repeat if needed.

Divisibility Rules Table Printable

Final Thoughts On Divisibility Rules

Divisibility rules are essential tools for quickly determining whether a number is divisible by another without performing long division. These rules can save you time when solving problems in math or when checking your work.

By mastering the divisibility rules for numbers like 2, 3, 4, 5, 6, 9, 10, 11, 12, and 13, you’ll become more efficient and confident in your mathematical abilities.

Remember to practice these rules often to solidify your understanding and improve your mental math skills.