Divisibility Rule of 9
The divisibility rule of 9 states that if the sum of digits of any number is divisible by 9, then the number is also divisible by 9. It helps us in various concepts like finding divisors, HCF, LCM, measurements, and division. Divisibility by 9 is a rule that allows us to find whether a number is divisible by 9 or not without performing long division.
1.  What is the Divisibility Rule of 9? 
2.  Divisibility Rule of 9 for Large Numbers 
3.  Divisibility by 3 and 9 
4.  Divisibility Rule of 9 and 11 
5.  FAQs on Divisibility Rule of 9 
What is the Divisibility Rule of 9?
The divisibility rule of 9 helps us to find whether a number is a multiple of 9 or not without performing the actual division. Some of the multiples of 9 are 9, 18, 27, 36, 45, etc. Do you see a common pattern in the sum of the digits of these numbers? The sum of digits of all these numbers is itself a multiple of 9. For example, 18 is 1+8 = 9, which is divisible by 9, 27 is 2+7 = 9, which is divisible by 9, etc. So, as per the divisibility test of 9, if the sum of all the digits of a number is a multiple of 9, then the number is also divisible by 9.
Divisibility Rule of 9 Fun Activity
There is a fun activity based on the divisibility rule of 9. Ask your friend to think of any singledigit nonzero number. Ask her/him to take three times that number, make sure you should not know what is the number picked up by her/him. Now, tell her/him to multiply the result by 3. Now, ask her/him how many digits are there in the answer and tell you one of the digits from the resultant value. Then you can find out what the other digit of the number is, by using the divisibility test of 9. The other digit can be obtained by subtracting the known digit from 9. Let us try it out with a number 6. Three times 6 is 18. Now, multiply 18 by 3, which is 54. If we know any one of the digits, let's say 4, we can easily find out what is the other digit by subtracting it from 9, i.e., 9  4 = 5. So, the other digit is 5.
Divisibility Rule of 9 for Large Numbers
The divisibility rule of 9 remains the same for large numbers. The only difference is that we use the divisibility test of 9 repeatedly until we get the sum of the digits of the number closer to 9. For example, to find if 2374878 is divisible by 9 or not, we first find the sum of the digits, which is 2+3+7+4+8+7+8 = 39. Now, we will again add 3 and 9, which is 3+9 = 12, and 12 is not divisible by 9. So, 2374878 is not divisible by 9. Let us take another example. To find if 456318 is divisible by 9 or not, we first find the sum of the digits, which is 4+5+6+3+1+8 = 27. Now, we will again add 2 and 7, which is 2+7 = 9, and 9 is divisible by 9. So, 456318 is divisible by 9. Let us look at the steps to apply the divisibility rule of 9 easily with any large or smaller numbers:
 Step 1: Find the sum of all the digits of the given number.
 Step 2: Check if the sum is divisible by 9 or not. If it is still a large number, add the digits again.
 Step 3: Check if the new sum is divisible by 9 or not. Repeat this process if you still find it difficult to figure out whether the sum of the digits is divisible by 9 or not.
 Step 4: If the final sum is divisible by 9, then the original number would also be divisible by 9.
This is how the divisibility test of 9 works.
Divisibility by 3 and 9
Both the divisibility test of 9 and 3 are based on the same principle, which states that the sum of the digits of the given number should be divisible by them. To check if a number is divisible by 3 or not, the sum of all the digits of the number should be divisible by 3, while on the other hand in the case of divisibility rule by 9, if the sum of all the digits of the number is divisible by 9, then the number is also a multiple of 9.
For example, to find whether 459072 is divisible by 9 and 3 or not, let us find the sum of the digits. The sum is 4+5+9+0+7+2 = 27, which can be again summed to 2+7 = 9. The sum '9' is divisible by both 9 and 3, therefore, 459072 is divisible by both 9 and 3. Here, one important fact is that every number which is divisible by 9 is also divisible by 3 because 9 is itself a multiple of 3. On the other hand, every number which is divisible by 3 may or may not be divisible by 9.
Divisibility Rule of 9 and 11
The divisibility rule of 9 and 11 is different. As discussed earlier, the divisibility test of 9 says that the sum of the digits of the given number should be divisible by 9. However, the divisibility rule of 11 states that a number is divisible by 11 if the difference of the sum of the digits at even places and odd places is 0 or divisible by 11. So, we first find the difference of the sum of digits at even places and at odd places. It should be noted that both the rules are based on the sum of digits, but in the case of 11, we have to find the sum of digits at odd place values and at even place values separately, and then if the difference between the two sums is divisible by 11, the number will also be divisible by 11.
For example, let us find whether 99990 is divisible by 9 and 11 or not. The sum of all the digits is 9+9+9+9+0 = 36, which is divisible by 9, so 99990 is divisible by 9. Now, let us find the sum of the digits at even place values starting from the right, 9+9 = 18. The sum of digits at odd places from the right side is 0+9+9 = 18. Now, the difference between the two is 1818 = 0, which is divisible by 11 (as 0 is divisible by every number). So, 99990 is divisible by both 9 and 11.
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Divisibility Rule of 9 with Examples

Example 1: Using the divisibility rule of 9, state whether 724 is divisible by 9 or not.
Solution: Let us find the sum of all the digits of the number 724.
7+2+4 = 13
Here, 13 is not divisible by 9, so as per the divisibility test of 9, we can say that 724 is not divisible by 9.

Example 2: Check the divisibility of the following number by 9: 18972 without performing long division.
Solution: We can use the divisibility test of 9 here which states that if the sum of all the digits of a number is divisible by 9, then the number is also divisible by 9. The sum of digits of 18972 is 1+8+9+7+2 = 27, which is divisible by 9. Therefore, we can say that 18972 is divisible by 9.

Example 3: Find the smallest 3digit number divisible by 9.
Solution: The smallest 3digit number is 100. In order to check the smallest number of 3 digits which can be a multiple of 9, we need to find the sum of the digits. Let us consider the units place as blank, so we have 10_. Now we have to find a digit that could come in the blank such that the sum of 1 and that digit is 9. That digit is 8, as 1+8 = 9. Therefore, 108 is the smallest 3digit number which is divisible by 9.
FAQs on Divisibility Rule of 9
What is the Divisibility Rule of 9?
The divisibility rule of 9 states that if the sum of all the digits of a number is divisible by 9, then the number would be divisible by 9. It helps us to find whether 9 is a factor of any number or not without performing the actual division. For example, let us check if 85304 is divisible by 9. Since 8 + 5 + 3 + 0 + 4 = 20 and 20 is not divisible by 9, it can be said that 85304 is not divisible by 9.
What do the Divisibility Rules for 9 and 3 have in Common?
The divisibility test of 9 and 3 are based on the sum of the digits of the number. If the sum of the digits of the given number is divisible by 9 and 3, then the number will be divisible by 9 and 3 respectively. Note that all the numbers that are divisible by 9 are also divisible by 3, as 9 is itself a multiple of 3.
Using the Divisibility Rule of 9, Check if 1450 is Divisible by 9?
The sum of all the digits of 1450 is 1+4+5+0 = 10, which is not divisible by 9. So, 1450 is not divisible by 9, as per the divisibility test for 9.
How do you know if a Big Number is Divisible by 9?
With large numbers, we repeat the process of adding the digits of a number if we are not sure whether the sum of the digits is divisible by 9 or not. If that sum is divisible by 9, then the number is also divisible by 9. For example, to check whether 5409279 is divisible by 9 or not, we add all the digits, 5 + 4 + 0 + 9 + 2 + 7 + 9 = 36, which can be further added as 3 + 6 = 9, and 9 is divisible by 9. So, 5409279 is divisible by 9.
Using the Divisibility Rule of 9, Check if 8955 is Divisible by 9?
The sum of all the digits of 8955 is 8+9+5+5 = 27, which is divisible by 9. So, 8955 is divisible by 9, as per the divisibility rule for 9.
What is the Divisibility Rule of 3 and 9?
The divisibility rule of 3 states that if the sum of the digits of the given number is divisible by 3 then the number is divisible by 3. For example, let us check if 632 is divisible by 3. Since 6 + 3 + 2 = 11, and 11 is not divisible by 3, we can say that 632 is not divisible by 3. The divisibility rule of 9 states that if the sum of the digits of the given number is divisible by 9 then the number is divisible by 9. For example, let us check if 5499 is divisible by 9. Since 5 + 4 + 9 + 9 = 27, and 27 is divisible by 9, we can say that 5499 is divisible by 9.
Test Whether 9846 is Divisible by 9.
Using the divisibility rule of 9, let us check if 9846 is divisible by 9 or not. Since 9 + 8 + 4 + 6 = 27 and 27 is divisible by 9, we can say that 9846 is divisible by 9.
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