 # Number Theory Properties of Divisibility | Basic Divisibility Rules with Examples

Division belongs to the arithmetic operation. So, the students who wish to become a master in math are suggested to know more about the division and divisibility rules. Divisibility is the basic part of number theory. Let us say a divides b if be leaves a remainder of 0 when divided by a. We say that b is divisible by a. Here a is the divisor and b is multiple. It is denoted as a/b. The notations a/b may also apply to negative integers a and b where q is a negative integer or when a and b are both negative.

## Basic Properties of Divisibility | Divisibility Rules

Here is the list of properties of divisibility that help you solve the problems on divisibility quite easily. They are explained in detail as below
Property 1:
When a number is divisible by another number, it is also divisible by the factors of the number.
Example:
12 is divisible by 6
Then 12 is also divisible by 2 and 3 which are the factors of 6.
Property 2:
When a number is divisible by two or more co-prime numbers, it is also divisible by their product.
Example:
18 is divisible by both 2 and 3 which are co-primes.
Then, 18 is also divisible by 6 which is the product of 2 and 3.
Property 3:
When a number is a factor of two given numbers, it is also a factor of their sum and difference.
Example:
2 is a factor 4 and 8.
Then,
2 is also a factor of 12(4 + 8).
2 is also a factor of 4(8 – 4).
Property 4:
When a number is a factor of another number, it is also a factor of any multiple of that number.
Example:
6 is a factor of 18.
Then,
6 is also a factor of 24 = (4 × 6)
6 is also a factor of 36 = (6 × 6)
Property 5:
If an integer is divisible by two or more different numbers, then is it also divisible by the least common multiple of those numbers.
Example:
24 is divisible by both 2 and 3.
24 is also divisible by 6 which is the least common multiple of 2 and 3.

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### Divisibility Properties Examples

Example 1.
How many numbers from 0 to 10 are exactly divisible by both 2 and 3?
Solution:
The numbers from 0 to 10 are
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Numbers which are divisible by both 2 and 3 are also divisible by the least common multiple of 2 and 3.
Least common multiple of 2 and 3 is 6.
Therefore from 0 to 10, there are 1 number that is divisible by 2 and 3.

Example 2.
Check whether 52563744 is divisible by 3.
Solution:
According to the divisibility rule for 3, if the sum of all the digits is divisible by 3 or a multiple of 3, then the number is divisible by 3.
Add all the digits in the number 52563744
5 + 2 + 5 + 6 + 3 + 7 + 4 + 4 = 36
The sum of the digits in the given number is 36 which is a multiple of 3.
So, 52563744 is divisible by 3.

Example 3.
Check if 525 is divisible by 5.
Solution:
According to divisibility rule for 5, if the last digit contains 0 or 5 it will be divisible by 5.
The number 525 has 5 in its last digit.
So, 525 is divisible by 5.

Example 4.
Check if 744 is divisible by 8.
Solution:
According to the test of divisibility for 8, in a number, if the number formed by the last 3 digits is divisible by 8, then the number is divisible by 8.
744 is divisible by 8.

Example 5.
Check if 626 is divisible by 2.
Solution:
According to the divisibility rule for 2, if the last digit has 0, 2, 4, 6, 8  then the number is divisible by 2.
626 has 6 in its last digit.
So, the number 626 is divisible by 2.

Students who want to know more about divisibility can make use of 5th Grade math. This page contains worksheets, practice questions, and word problems on specific topics.

Read More Divisibility Rules of Numbers 2-10:

 Divisible by 2 Divisible by 3 Divisible by 4 Divisible by 5 Divisible by 6 Divisible by 7 Divisible by 8 Divisible by 9 Divisible by 10

### FAQs on Divisibility Rules

1. Is zero divisible by any number?

Zero is divisible by any number except by itself.

2. What is the 8 divisibility rule?

If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.

3. What is the divisibility rule of 1 to 10?

A number is divisible by 10, if last digit is 0.
Example: 20 is divisible by 10 as last digit is 0. 43 is not divisible by 10 as last digit is not 0.

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