How to Tell if a Number Is Divisible by 9
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 Rule of 9 and 3 |
| 4. | Divisibility Test of 9 and 11 |
| 5. | Divisibility Rule of 9 Examples |
| 6. | 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 single-digit non-zero 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's try it out with a number, let's say 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
With large numbers also, the rule is the same. 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 Rule of 9 and 3
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 Test of 9 and 11
We have already discussed the divisibility rule of 9, so here let's understand the divisibility by 11. It is by finding the difference of the sum of digits at even places and at odd places. 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 18-18 = 0, which is divisible by 11 (as 0 is divisible by every number). So, 99990 is divisible by both 9 and 11.
Articles Related to Divisibility Test of 9
Also, check the given articles based on the divisibility test of different numbers.
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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 by 9, we can say that 724 is also not divisible by 9.
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Example 2: Check the divisibility of 18972 by 9, 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.
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Example 3: Find the smallest 3-digit number divisible by 9.
Solution: The smallest 3-digit number is 100. But to check which smallest number of 3 digits can be a multiple of 9, we have to find the sum of the digits. Let us consider 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 3-digit number which is divisible by 9.
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FAQs on Divisibility Rule of 9
What is the Divisibility Rule of 9?
The divisibility rule of 9 states that when the sum of all the digits of a number is divisible by 9, then only 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.
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 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 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.
How to Tell if a Number Is Divisible by 9
Source: https://www.cuemath.com/numbers/divisibility-rule-of-9/#:~:text=The%20sum%20of%20digits%20of%20all%20these%20numbers%20is%20itself,is%20also%20divisible%20by%209.