Divisibility Rule of 9: Methods, Examples

What is the Divisibility Rule of 9?

Divisibility rule of 9 is a simple method to check if a number is exactly divisible by 9 without doing long division. According to this rule, if we add up all the digits of a number and the total is divisible by 9, then the original number is also divisible by 9.

Understanding what is the divisibility rule of 9 makes it easy to identify multiples of 9. Numbers like 18, 27, 45, and 81 all follow this rule because the sum of their digits is always a multiple of 9. The rule for divisibility by 9 is also very important in learning topics such as factors, multiples, HCF and LCM in maths.

Read More: How to Do Long Division

Divisibility Rule of 9 for Large Numbers

Divisibility rule of 9 stays the same even for the larger numbers. But for larger numbers, we need to apply the rule step by step until the sum of the digits becomes easy to check.

To make the rule for divisibility by 9 for larger numbers easy to understand, here is the step-by-step explanation:

• Step 1: Add all the digits of the number.

• Step 2: If the sum is large, add the digits again.

• Step 3: Repeat the process until you get a single-digit or smaller number.

• Step 4: If the final sum is divisible by 9, then the original number is divisible by 9.

For example, to check if 2,374,878 is divisible by 9. First, add the digits: 2 + 3 + 7 + 4 + 8 + 7 + 8 = 39. Since 39 is still a two-digit number, we add again: 3 + 9 = 12. Now, 12 is not divisible by 9, which means 2,374,878 is not divisible by 9.

Read More: Division Questions for Class 4

Divisibility by 3 and 9

Divisibility by 3 and 9 rules are very similar to each other because both are based on the sum of the digits of a number. To check divisibility by 3, we add all the digits and see if the sum is divisible by 3. For the divisibility rule of 9, we do the same, but the final sum should be divisible by 9 instead.

For example, to check if 459072 is divisible by 3 and 9. First, add the digits: 4 + 5 + 9 + 0 + 7 + 2 = 27. Add again: 2 + 7 = 9. Since 9 is divisible by both 3 and 9, the number 459072 is divisible by both.

It is important for students to remember that every number divisible by 9 will also be divisible by 3, because 9 is a multiple of 3. However, a number divisible by 3 may not always be divisible by 9. For example, 24 is divisible by 3, but it is not divisible by 9.

Read More: Multiplication Tables

Divisibility Rule of 9 and 11

Divisibility rule of 9 and 11 is different from each other. For 9, we just add all the digits of a number. If the total is divisible by 9, then the number is also divisible by 9. But as per the divisibility rule of 11, we add the digits at odd places and even places separately, then find the difference of these two sums. If the difference is 0 or divisible by 11, then the number is divisible by 11.

For example, if we check the number 99990 for divisibility by 9 and 11, we first need to add all digits: 9 + 9 + 9 + 9 + 0 = 36. Since 36 is divisible by 9, the number is divisible by 9. Now for 11, first add digits at even places: 9 + 9 = 18. Then, add digits at odd places: 0 + 9 + 9 = 18. The difference is 18 – 18 = 0. Since 0 is divisible by 11, the number is divisible by both 9 and 11.

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Divisibility Rule of 9 with Examples

The divisibility rule of 9 helps us to check whether a number is a multiple of 9 without doing long division. Check the divisibility rule of 9 with examples given below to understand it better:

Example 1: Check if 724 is divisible by 9.

Solution:

Add the digits of the number: 7 + 2 + 4 = 13.

Since 13 is not divisible by 9, the number 724 is not divisible by 9.

Example 2: Use the rule for divisibility by 9 to check if 18972 is divisible by 9.

Solution:

Add the digits of the number: 1 + 8 + 9 + 7 + 2 = 27.

Since 27 is divisible by 9, the number 18972 is divisible by 9.

Example 3: Apply the divisibility rule of 9 to 3526.

Solution:

Add all the digits of the number: 3 + 5 + 2 + 6 = 16.

Since 16 is not divisible by 9, the number 3526 is not divisible by 9.

Also Read: Math Word Problems: Addition, Subtraction, Multiplication, Division

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