Divisibility Rules

Divisibility rules are shortcuts that allow you to quickly determine if a number is divisible by another number without having to perform long division. Each divisibility rule is based on the properties of the divisors and the digits of the dividend.

Divisibility Rule for 2

A number is divisible by 2 if its last digit is even.

Examples:

• 12 is divisible by 2 because its last digit is even (2).
• 23 is not divisible by 2 because its last digit is odd (3).

Divisibility Rule for 3

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

Examples:

• 123 is divisible by 3 because the sum of its digits is 1 + 2 + 3 = 6, which is divisible by 3.
• 456 is not divisible by 3 because the sum of its digits is 4 + 5 + 6 = 15, which is not divisible by 3.

Divisibility Rule for 4

A number is divisible by 4 if the last two digits are divisible by 4.

Examples:

• 1234 is divisible by 4 because the last two digits (34) are divisible by 4.
• 5678 is not divisible by 4 because the last two digits (78) are not divisible by 4.

Divisibility Rule for 5

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

Examples:

• 125 is divisible by 5 because its last digit is 5.
• 230 is divisible by 5 because its last digit is 0.
• 341 is not divisible by 5 because its last digit is not 0 or 5.

Divisibility Rule for 6

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

Examples:

• 1236 is divisible by 6 because it is divisible by both 2 (its last digit is even) and 3 (the sum of its digits is 1 + 2 + 3 + 6 = 12, which is divisible by 3).
• 4567 is not divisible by 6 because it is not divisible by 2 (its last digit is odd).

Divisibility Rule for 9

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

Examples:

• 12345 is divisible by 9 because the sum of its digits is 1 + 2 + 3 + 4 + 5 = 15, which is divisible by 9.
• 67890 is not divisible by 9 because the sum of its digits is 6 + 7 + 8 + 9 + 0 = 30, which is not divisible by 9.

Divisibility Rule for 10

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

Examples:

• 1230 is divisible by 10 because its last digit is 0.
• 4567 is not divisible by 10 because its last digit is not 0.

Divisibility Rule for 11

A number is divisible by 11 if the alternating sum of its digits is divisible by 11. The alternating sum is calculated by subtracting the sum of the digits in the odd positions from the sum of the digits in the even positions.

Examples:

• 12345 is divisible by 11 because the alternating sum of its digits is (1 - 2 + 3 - 4 + 5) = 3, which is divisible by 11.
• 67890 is not divisible by 11 because the alternating sum of its digits is (6 - 7 + 8 - 9 + 0) = -2, which is not divisible by 11.

Divisibility Rule for 12

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

Examples:

• 1236 is divisible by 12 because it is divisible by both 3 (the sum of its digits is 1 + 2 + 3 + 6 = 12, which is divisible by 3) and 4 (the last two digits are 36, which is divisible by 4).
• 4567 is not divisible by 12 because it is not divisible by 4 (the last two digits are 67, which is not divisible by 4).

Divisibility Rules From 1 to 13

Divisibility Rule for 1

• A number is divisible by 1 if and only if it is not zero.

Divisibility Rule for 2

• A number is divisible by 2 if and only if its last digit is even.

Divisibility Rule for 3

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

Divisibility Rule for 4

• A number is divisible by 4 if and only if the last two digits are divisible by 4.

Divisibility Rule for 5

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

Divisibility Rule for 6

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

Divisibility Rule for 7

• There is no simple divisibility rule for 7. However, there are several tests that can be used to determine if a number is divisible by 7.

Divisibility Rule for 8

• A number is divisible by 8 if and only if the last three digits are divisible by 8.

Divisibility Rule for 9

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

Divisibility Rule for 10

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

Divisibility Rule for 11

• A number is divisible by 11 if and only if the alternating sum of its digits is divisible by 11. The alternating sum is found by subtracting the sum of the digits in the odd positions from the sum of the digits in the even positions.

Divisibility Rule for 12

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

Divisibility Rule for 13

• There is no simple divisibility rule for 13. However, there are several tests that can be used to determine if a number is divisible by 13.

Divisibility Rule for Prime Numbers

Prime numbers are whole numbers greater than 1 whose only factors are 1 and themselves. There are many divisibility rules that can be used to determine if a number is divisible by a given prime number.

Divisibility Rule for 2

A number is divisible by 2 if its last digit is even.

Divisibility Rule for 3

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

Divisibility Rule for 5

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

Divisibility Rule for 7

There is no simple divisibility rule for 7. However, there are a few tricks that can be used to determine if a number is divisible by 7.

• Double the last digit and subtract it from the rest of the number. If the result is divisible by 7, then the original number is divisible by 7.
• For example, to determine if 123 is divisible by 7, we would double the last digit (3) and subtract it from the rest of the number (12): 12 - 6 = 6. Since 6 is divisible by 7, we know that 123 is also divisible by 7.

Divisibility Rule for 11

There is no simple divisibility rule for 11. However, there are a few tricks that can be used to determine if a number is divisible by 11.

• Add the digits in the odd positions (1, 3, 5, etc.) and then add the digits in the even positions (2, 4, 6, etc.). If the difference between these two sums is divisible by 11, then the original number is divisible by 11.
• For example, to determine if 12345 is divisible by 11, we would add the digits in the odd positions (1 + 3 + 5 = 9) and then add the digits in the even positions (2 + 4 = 6). The difference between these two sums is 3, which is divisible by 11. Therefore, we know that 12345 is also divisible by 11.

Divisibility Rule for 13

There is no simple divisibility rule for 13. However, there are a few tricks that can be used to determine if a number is divisible by 13.

• Multiply the last digit by 4 and add it to the rest of the number. If the result is divisible by 13, then the original number is divisible by 13.
• For example, to determine if 123 is divisible by 13, we would multiply the last digit (3) by 4 and add it to the rest of the number (12): 12 + 12 = 24. Since 24 is divisible by 13, we know that 123 is also divisible by 13.

Divisibility Rule for 17

There is no simple divisibility rule for 17. However, there are a few tricks that can be used to determine if a number is divisible by 17.

• Multiply the last digit by 5 and add it to the rest of the number. If the result is divisible by 17, then the original number is divisible by 17.
• For example, to determine if 123 is divisible by 17, we would multiply the last digit (3) by 5 and add it to the rest of the number (12): 12 + 15 = 27. Since 27 is divisible by 17, we know that 123 is also divisible by 17.

Divisibility Rule for 19

There is no simple divisibility rule for 19. However, there are a few tricks that can be used to determine if a number is divisible by 19.

• Multiply the last digit by 2 and add it to the rest of the number. If the result is divisible by 19, then the original number is divisible by 19.
• For example, to determine if 123 is divisible by 19, we would multiply the last digit (3) by 2 and add it to the rest of the number (12): 12 + 6 = 18. Since 18 is divisible by 19, we know that 123 is also divisible by 19.

Divisibility Rules Chart

Divisibility rules are simple tests that can be used to determine if a number is divisible by another number without performing long division. These rules are based on the properties of the divisibility of numbers and can be used to quickly and easily determine if a number is divisible by a given divisor.

Divisibility Rules for Common Divisors

Examples of Using Divisibility Rules

• To determine if the number 1234 is divisible by 2, we look at the last digit of the number. Since the last digit is even, we know that 1234 is divisible by 2.
• To determine if the number 456 is divisible by 3, we add the digits of the number: 4 + 5 + 6 = 15. Since 15 is divisible by 3, we know that 456 is divisible by 3.
• To determine if the number 8765 is divisible by 4, we look at the last two digits of the number. Since the last two digits are divisible by 4, we know that 8765 is divisible by 4.
• To determine if the number 98765 is divisible by 5, we look at the last digit of the number. Since the last digit is not 0 or 5, we know that 98765 is not divisible by 5.

Conclusion

Divisibility rules are a useful tool for quickly and easily determining if a number is divisible by another number. These rules can be used to simplify calculations and to solve problems involving divisibility.

Solved Examples of Divisibility Rules

Divisibility by 2

Example 1: Determine if 1234 is divisible by 2.

Solution:

• The last digit of 1234 is 4, which is even.
• Therefore, 1234 is divisible by 2.

Example 2: Determine if 56789 is divisible by 2.

Solution:

• The last digit of 56789 is 9, which is odd.
• Therefore, 56789 is not divisible by 2.

Divisibility by 3

Example 1: Determine if 1234 is divisible by 3.

Solution:

• Add the digits of 1234: 1 + 2 + 3 + 4 = 10.
• 10 is not divisible by 3.
• Therefore, 1234 is not divisible by 3.

Example 2: Determine if 56789 is divisible by 3.

Solution:

• Add the digits of 56789: 5 + 6 + 7 + 8 + 9 = 35.
• 35 is divisible by 3.
• Therefore, 56789 is divisible by 3.

Divisibility by 4

Example 1: Determine if 1234 is divisible by 4.

Solution:

• Consider the last two digits of 1234: 34.
• 34 is not divisible by 4.
• Therefore, 1234 is not divisible by 4.

Example 2: Determine if 56789 is divisible by 4.

Solution:

• Consider the last two digits of 56789: 89.
• 89 is not divisible by 4.
• Therefore, 56789 is not divisible by 4.

Divisibility by 5

Example 1: Determine if 1234 is divisible by 5.

Solution:

• The last digit of 1234 is 4, which is not 0 or 5.
• Therefore, 1234 is not divisible by 5.

Example 2: Determine if 56789 is divisible by 5.

Solution:

• The last digit of 56789 is 9, which is not 0 or 5.
• Therefore, 56789 is not divisible by 5.

Divisibility by 6

Example 1: Determine if 1234 is divisible by 6.

Solution:

• 1234 is divisible by 2 (last digit is even) and 3 (sum of digits is divisible by 3).
• Therefore, 1234 is divisible by 6.

Example 2: Determine if 56789 is divisible by 6.

Solution:

• 56789 is divisible by 3 (sum of digits is divisible by 3) but not by 2 (last digit is odd).
• Therefore, 56789 is not divisible by 6.

Divisibility by 9

Example 1: Determine if 1234 is divisible by 9.

Solution:

• Add the digits of 1234: 1 + 2 + 3 + 4 = 10.
• 10 is not divisible by 9.
• Therefore, 1234 is not divisible by 9.

Example 2: Determine if 56789 is divisible by 9.

Solution:

• Add the digits of 56789: 5 + 6 + 7 + 8 + 9 = 35.
• 35 is divisible by 9.
• Therefore, 56789 is divisible by 9.

Divisibility by 10

Example 1: Determine if 1234 is divisible by 10.

Solution:

• The last digit of 1234 is 4, which is not 0.
• Therefore, 1234 is not divisible by 10.

Example 2: Determine if 56789 is divisible by 10.

Solution:

• The last digit of 56789 is 9, which is not 0.
• Therefore, 56789 is not divisible by 10.

Divisibility Rules FAQs

What is a divisibility rule?

A divisibility rule is a test that can be applied to a number to determine whether it is divisible by another number without performing long division.

What are some common divisibility rules?

Some common divisibility rules include:

• Rule for 2: A number is divisible by 2 if its last digit is even.
• Rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Rule for 4: A number is divisible by 4 if the last two digits are divisible by 4.
• Rule for 5: A number is divisible by 5 if its last digit is 0 or 5.
• Rule for 6: A number is divisible by 6 if it is divisible by both 2 and 3.
• Rule for 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
• Rule for 10: A number is divisible by 10 if its last digit is 0.

How can I use divisibility rules?

Divisibility rules can be used to quickly and easily determine whether a number is divisible by another number. This can be useful for a variety of purposes, such as:

• Simplifying fractions
• Finding common factors
• Determining the greatest common divisor (GCD) of two numbers
• Finding the least common multiple (LCM) of two numbers

Are there any divisibility rules for prime numbers?

There are no divisibility rules for prime numbers. A prime number is a number that is only divisible by 1 and itself.

What is the most important divisibility rule?

The most important divisibility rule is the rule for 10. This rule can be used to quickly and easily determine whether a number is divisible by 10, which is useful for a variety of purposes, such as rounding numbers and converting between different units of measurement.