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The divisibility rule of 9 is used most frequently in the world of mathematics to determine the divisibility of complex numbers by 9. The rule of 9 for divisibility is a simple yet effective method in math to figure out if a number is divisible by 9. This principle can be applied in mental arithmetic, number theory, and various real-life scenarios. In this article, we will investigate the functioning of the rule, review solved examples, and talk about its importance.

## Divisibility Rule of 9

**The divisibility rule of 9 states that if the total of the digits of a number can be divided evenly by 9, then the number itself can also be divided by 9.** Knowing the divisibility rule for 9 assists us in different concepts such as determining divisors, HCF, LCM, measuring, and dividing.

The divisibility rule for 9 helps us determine if a number can be divided by 9 without needing to do long division. You must have noticed the the 9 divisibility rule is similar to that of 3.

## Divisibility Rule for 9

The rule for determining if a number is divisible by 9 allows us to check without having to do long division. Certain numbers such as 9, 18, 27, 36, 45, etc., are multiples of 9. Can you identify a recurring pattern in the total of their individual digits?

The total of all the digits in these numbers is a multiple of 9. If we take 18: 1+8 = 9, divisible by 9, and 27: 2+7 = 9, divisible by 9, then according to the divisibility rule of 9, a number is divisible by 9 if the sum of its digits is a multiple of 9.

## Divisibility by 9

A number is said to be divisible by a another number if it leaves the remainder 0 and gives whole number as a quotient. To find if a number is divisible by a certain other number, we need to perform division. Division is a time consuming process, especially in case of large numbers. Can’t we find if a number is divisible by another number using shortcuts. This is where the divisibility rule comes in.

The divisibility rules are used to validate if a number is actually divisible by a certain number or not. Though it doesn’t tell the value of quotient, but it helps us know if the number is completely divisible or not. The divisibility rule of 9 is quite simple and the concept of it is based on the rule for divisibility by 3.

## Why Divisibility Rule of 9 Functions Correctly

The reason why the rule is effective is because of the characteristics of the decimal number system. In modular arithmetic, the remainder of a number’s digits divided by 9 is equal to the number itself divided by 9. If the total of a number’s digits can be divided by 9, then the number itself can also be divided by 9. This characteristic stems from the way numbers are shown and operated on in the decimal system.

## Test of Divisibility of 9

Let us test the divisibility rule of 9 by going through some solved examples.

**Example 1: Check whether 87639 is divisible by 9?**

**Solution: **Given number = 87639

Sum of the digits of the given number = 8+7+6+3+9 = 33

As the sum of digits is 33, which is not divisible by 9. Hence the given number is not divisible by 9.

**Example 2: Is 786 divisible by 9**

**Solution: **To know, let us first calculate the sum of its digits

7+8+6 = 21, which is not divisible by 9

As the sum of digits is not divisible by 9, the number 786 will also be not divisible by 9

**Example 3: The given number is 34117326. Check if it is divisible by 9**

**Solution: **The sum of the given digits = 3+4+1+1+7+3+2+6 = 27

27 is divisible by 9

Hence the number 34117326 is also divisible by 9

**Example 4: Check the number 19997 for divisibility by 9**

**Solution: **The sum of digits of 19997 is 35

As 35 is not divisible by 9, so, 19997 is also not divisible by 9.

## Importance of Divisibility Rule of 9

Some of the importance of the divisibility rule for 9 is given hereunder.

- Mental Mathematics: The guideline makes calculations and verifications simpler, facilitating mental and efficient number manipulation.
- Error checking is beneficial for validating calculations, particularly in real-life situations such as accounting and inputting data.
- Understanding how to divide by 9 can be beneficial when solving problems in modular arithmetic and number theory.
- Identifying number patterns, like multiples of 9, can assist in solving puzzles and math problems.