Divisibility Rules

Divisibility rules are simple guidelines that help determine if one number can be evenly divided by another without performing the actual division. For instance, a number is divisible by 2 if its last digit is even, by 3 if the sum of its digits is divisible by 3, and by 5 if it ends in 0 or 5. These rules streamline calculations and are fundamental in various mathematical applications.

• 1. Divisibility Rule
• 2. Divisibility by 2 rule
• 3. Divisibility by 3 rule
• 4. Divisibility by 4 rule
• 5. Divisibility by 5 rule
• 6. Divisibility by 6 rule
• 7. Divisibility by 9 rule
• 8. Divisibility by 10 rule
• 9. Divisibility by 11 rule

1. Divisibility Rule :

These rules in mathematics are developed to make the decision to divide two numbers quickly. If we have to check whether a number is divisible by some other number, we can use these rules without performing a long division method.

It is an algorithm for determining whether a given number is divisible by a fixed divisor without performing the actual division.

For example, Let's suppose a girl has to distribute 108 chocolates among her 9 friends on her birthday. If she wonders if each one will get equal chocolates, she has to divide the number 108 by 9, but if that girl knows divisibility rules, she can easily tell whether or not she can divide the chocolates equally. Let’s check all important divisibility rules.

2. Divisibility by 2 Rule:

A number is only divided by 2 when the
last digit is even, i.e. (0, 2, 4, 6, 8).

$\bold{For \space Example:}$

• 4, 16, 20, 100, and 56 are divided by 2 because the last digit of all these numbers is even.
• 5, 13, 17, 21, and 77 are not divided by 2 because the last digit of all these numbers is not even.

3. Divisibility by 3 Rule:

A number is only divided by 3 when the
sum of all digits of that number is divided by 3.

$\bold{For \space Example:}$

• 87 is divided by 3 because the sum of the digits 8 + 7 = 15 is divided by 3.
• 458 is not divided by 3 because the sum of the digits 4 + 5 + 8 = 17 is not divided by 3.

4. Divisibility by 4 Rule:

A number is only divided by 4 when the
number formed by the last 2 digits is divided by 4.

$\bold{For \space Example:}$

• 1364 is divided by 4 because the number formed by the last 2 digits is 64, and that is divided by 4.
• 13641 is not divided by 4 because the number formed by the last 2 digits is 41, which is not divided by 4.

5. Divisibility by 5 Rule:

A number is only divided by 5 when the
last digit is 0 or 5.

$\bold{For \space Example:}$

• 123485, 280010, and 55555 are divided by 5 because the last digit of all these numbers is either 0 or 5.
• 5811, 2694, and 41 are not divided by 5 because the last digit of all these numbers is neither 0 nor 5.

6. Divisibility by 6 Rule:

A number is only divided by 6 when it
is divided by 2 and 3.

$\bold{For \space Example:}$

• 156 is divided by 6 because it is divided by 2 and 3 both.
• 456 is divided by 6 because it is divided by 2 and 3 both.
• 242 is not divided by 6 because it is divided by 2 but not by 2.
• 1007 is not divided by 6 because it is neither divided by 2 nor 3.

6. Divisibility by 7 Rule:

A number is only divided by 7 when it follows these steps

• Consider the last digit of the number and double it.
• Subtract the result from the remaining number.
• If the resulting number is 0 or a multiple of 7, then the original number is divisible by 7. Else, it is not divisible by 7.

$\bold{For \space Example:}$

• Consider the number 162453 If it is divisible by 7 or not.
• Consider the last digit, i.e., 7, and we have to double it, i.e., 7 x 2 = 14.

7. Divisibility by 9 Rule:

A number is only divided by 9 when the
sum of all digits of that number is divided by 9.

$\bold{For \space Example:}$

• 152829 is divided by 9 because the sum of all digits (1 + 5 + 2 + 8 + 2 + 9 = 27) of the number is divided by 9.
• 328151 is not divided by 9 because the sum of all digits (3 + 2 + 8 + 1 + 5 + 1 = 20) is not divided by 9.

8. Divisibility by 10 Rule:

A number is only divided by 10 when the
last digit of that number is 0.

$\bold{For \space Example:}$

• 450, 200, and 2010 are divided by 10 because the last digit of these numbers is 0.
• 158, 2269, and 43587 are not divided by 10 because the last digit of these numbers is not 0.

9. Divisibility by 11 Rule:

A number is only divided by 11 when the sum of the digits
in even places equals the sum of the digits in odd places.

$\bold{For \space Example:}$

• 121 is divided by 11 because the sum of digits in even places 2 equals the sum of digits at odd places (1 + 1 = 2).
• 5654 is divided by 11 because the sum of digits in even places (6 + 4 = 10) equals the sum of digits at odd places (5 + 5 = 10).
• 25631 is not divided by 11 because the sum of digits in even places (5 + 3 = 8) equals the sum of digits at odd places (2+6+1 = 9).

$\color{red} \bold{Related \space Pages:}$
Pre Algebra Calculators
Algebra Calculators
Vector Formula sheet