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 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 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 18-18 = 0, which is divisible by 11 (as 0 is divisible by every number). So, 99990 is divisible by both 9 and 11.

**☛ Related Topics**

- Divisibility Rule of 3
- Divisibility Rule of 4
- Divisibility Rule of 5
- Divisibility Rule of 6
- Divisibility Rule of 7
- Divisibility Rule of 8
- Divisibility Rule of 11
- Divisibility Rule of 13

## 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.

## FAQs

### What is an example of divisible by 9? ›

9, 18, 27, 36, 45, 63, 72, 81, 90 and 99.

**What is the divisibility test of 3 and 9? ›**

The divisibility rules for 3 and 9 are quite similar. As defined above, **if the sum of the digits of a number is a multiple of 3 or divisible by 3, then the number is divisible by 3**. Similarly, if the sum of the digits of a number is a multiple of 9 or divisible by 9, then the number is divisible by 9.

**What is divisibility rule Class 9? ›**

First, take any number (for this example it will be 492) and add together each digit in the number (4 + 9 + 2 = 15). Then take that sum (15) and determine if it is divisible by 3. **The original number is divisible by 3 (or 9) if and only if the sum of its digits is divisible by 3 (or 9)**.

**What is the divisible test of 9? ›**

Divisibility Rule of 9

That is, **if the sum of digits of the number is divisible by 9, then the number itself is divisible by 9**. Example: Consider 78532, as the sum of its digits (7+8+5+3+2) is 25, which is not divisible by 9, hence 78532 is not divisible by 9.

**How do you prove divisibility by 9 test? ›**

**A number is divisible by 9 if the sum of its digits is divisible by 9**. Proof: Consider a 3 -digit number. Now, the first term 9 ( 11 a + b ) is divisible by 9. If the second term ( a + b + c ) is divisible by 9, then the sum of the two terms is also divisible by 9, that is, the number a b c is divisible by 9.

**Are all multiples of 9 divisible by 3? ›**

Yes. Since 9 is divisible by 3, **all multiples of 9 are also divisible by 3**. Thus, all multiples of 9 are also multiples of 3. However, not all multiples of 3 are multiples of 9.

**Which is divisible by 9 is also divisible by *? ›**

So, any number that is divisible by 9 is also divisible by **3** because 3 is the factor of 9.

**Is 9 divisible by 3 yes or no? ›**

**9 is a multiple of 3**, therefore 1098 is divisible by 3.

**What is the rule of 9 in math? ›**

Rule 9. **If any two digits of a multi-digit number are interchanged and the smaller of the two numbers is subtracted from the larger, the result will always be a multiple of nine**.

**What is divisibility rule examples? ›**

Property 1: When a number is divisible by another number, it is also divisible by each of the factors of the number. Example: **36 is divisible by 12 and 2, 3 and 4 are the factors of 12**. ⇒ 36 is also divisible by 2, 3 and 4. i.e., 36 ÷ 2 = 18 , 36 ÷ 3 = 12 and.

### Is 36 divisible by 9? ›

**36 divided by 9 is 4**. This problem asks how many sets of nine go into 36.

**How do you find multiples of 9? ›**

Solution: We know that we can get multiples of a number by **multiplying it by 1, 2, 3**, …, and so on. So, the multiples of 9 are : 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, …

**What happens when you divide by 9? ›**

When you're dividing by 9, **you're dividing a number, or the dividend, equally into 9 groups**. You can do this by using place values, such as tens and ones, to organize your groups. To check your answer, or quotient, multiply the quotient by the divisor, or 9, and add the remainder.

**What are 9 factors? ›**

What are the total factors of 9? The factors of 9 are **1, 3 and 9**.

**Why does every multiple of 9 add up to 9? ›**

Consider any number with increasing digits; for example, 1256, 367, or 245,689. **Any such number, when multiplied by 9, has the property that the sum of the digits in the result must be exactly 9**. 9 x 1256 = 11,304, and 1 + 1 + 3 + 0 + 4 = 9. This general property is easy to prove.

**What is the least number divisible by 9? ›**

A number is divisible by 9, if and only if the sum of the digits of the number is divisible by 9. Complete step by step answer: The smallest number of 4 digits is **1000**.

**How many two digit numbers are there which are divisible by 9? ›**

Hence, there are **10** numbers of two digits which are divisible by 9.

**How to prove that a natural number is divisible by 9 if and only if its sum of digits is divisible by 9? ›**

Theorem. A number expressed in decimal notation is divisible by 9 if and only if the sum of its digits is divisible by 9. That is: **N=[a0a1a2…an]10=a0+a110+a2102+⋯+an10n is divisible by 9**.

**Is 108 divisible by 9? ›**

Playing with Numbers

Check the divisibility of the following number by 9. ∴ **108 is divisible by 9**. i.e. 13 is not divisible by 9.

**Are all numbers divisible by 3 also divisible by 9 Prove your answer by giving examples? ›**

**Every number divisible by 9 is divisible by 3**. For example, 7425 is divisible by 9, hence it is divisible by 3. However, a number divisible by 3 is not necessarily divisible by 9. For example 6, 12, 15, 21, 24, 30 are all divisible by 3 but none of them is divisible by 9.

### What is the multiple of 9 answer? ›

The first ten multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90.

**What is the first multiple of 9? ›**

The first 10 multiples of 9 are: 9, 18, 27, 36, 45, 54, 63, 72, 81, and 90.

**What number is divisible by 3 4 9? ›**

The LCM of 3, 4, and 9 is **36**.

36 . . . . ; multiples of 9 = 9, 18, 27, 36 . . . .) and choose the smallest multiple that is exactly divisible by 3, 4, and 9, i.e., 36.

**Is the number 152875 divisible by 9 true false? ›**

Example 5: Check the divisibility of 152875 by 9. Solution: The sum of the digits of 152875 is 1 + 5 + 2 + 8 + 7 + 5 = 28. This number is **notdivisible by 9**.

**Which number is divisible by all 1 to 9? ›**

That number is 2520, and **there is no number smaller that is divisible by all the integers 1 - 9**. Below is a breakdown: 2520 divided by 2520 equals 1.

**What number is divisible by both 2 and 9? ›**

Hence the smallest number divisible by numbers 2 to 9=**2520**.

**Is 18 divisible by 9 yes or no? ›**

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.

**What is divisible by 4 and 9? ›**

Ans. **2, 3,4,6,9,12,18,36**.

**Are 9 and 16 divisible? ›**

FAQs on LCM of 9 and 16

**The LCM of 9 and 16 is 144**. To find the LCM of 9 and 16, we need to find the multiples of 9 and 16 (multiples of 9 = 9, 18, 27, 36 . . . . 144; multiples of 16 = 16, 32, 48, 64 . . . . 144) and choose the smallest multiple that is exactly divisible by 9 and 16, i.e., 144.

**What is an example of the rule of 9? ›**

The front and back of the head and neck equal 9% of the body's surface area. The front and back of each arm and hand equal 9% of the body's surface area. The chest equals 9% and the stomach equals 9% of the body's surface area. The upper back equals 9% and the lower back equals 9% of the body's surface area.

### What is the rule of 9 also called? ›

The Rule of Nines, also known as the **Wallace Rule of Nines**, is a tool used by trauma and emergency medicine providers to assess the total body surface area (TBSA) involved in burn patients.

**What is an example of test of divisibility in real life? ›**

Divisibility Defined

For example, **when you split a sandwich with your brother, share a pack of gum, or make sure that you and your cousin each have the same number of french fries without any extra left over**, you are working with a divisible number of items.

**What are examples of divisible? ›**

That is, any number that ends with 2, 4, 6, 8, or 0 will give 0 as the remainder when divided by 2. For example, **12, 46, and 780 are all divisible by 2**. If the sum of the digits of a number is divisible by 3, then the number as a whole would also be divisible by 3. For example, take the number 753.

**What numbers between 100 and 200 are divisible by 9? ›**

Solution: (i) Integers between 100 and 200, that are divisible by 9 are **108,117,126,…,198**.

**Is 24 divisible by 9? ›**

**If the sum of the digits is not divisible by 9, then 24 is not divisible by 9**. 6 is not divisible by 9, Therefore, 24 is not divisible by 9 and the answer to the question, "Is 24 Divisible By 9?" is No.

**How many 9 are between 1 and 100? ›**

There are total **20 nines** between 1–100.

**How many 9s are there in 90? ›**

Nine 10s made 90. The missing number is nine.

**What are the factor pairs of 9? ›**

The factors of 9 are 1, 3 and 9.

**What is the strategy for dividing by 9? ›**

Divide by 9 - Similar to the divide by 3 rule, **if the sum of all the digits is divisible by 9, then the entire number is divisible by 9**. For example, we know that 18332145 is divisible by 9 because 1+8+3+3+2+1+4+5 = 27 and 27 ÷ 9 = 3. Divide by 10 - If the number ends in a 0, then it is divisible by 10.

**How many times does 9 divide 45? ›**

45 divided by 9 is equal to **5**.

### How do you divide 9 11? ›

When we divide 9 by 11, **the result is a repeating decimal: 0.818181818181**. We can round the repeating decimal so that the answer is 0.82. See full answer below.

**What two numbers make 9? ›**

...

**Important Notes:**

- Factors of 9 are 1, 3, and 9.
- 1 is a universal factor.
- Factors of a number are the numbers we multiply to get that particular number.

**Is 9 a factor of 36 yes or no? ›**

The factors of 36 are the numbers that divide 36 exactly without leaving any remainder. Thus, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

**Is 9 a factor of 32 yes or no? ›**

So in general we can say that there are 6 positive and 6 negative common factors of 32. Other Integer Numbers like 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, cannot be the factors of 32 as they leaves remainder when they divide 32.

**What are the divisible factors of 9? ›**

When we divide 9 by 3, it is exactly divisible and leaves no remainder which means that 3 is a factor of 9. Note that 1 and the number itself are always factors of the number. Thus, the factors of 9 are **1, 3, and 9**.

**Is 21 divisible by 9? ›**

Since **21 is not divisible by 9**, therefore 53247 is not divisible by 9. For a number to be divisible by 9 sum of digits must be divisible by 9. Since 27 is divisible by 9, therefore 4968 is divisible by 9. For a number to be divisible by 9 sum of digits must be divisible by 9.

**What is the rule for the multiples of 9? ›**

In other words, the multiples of 9 are the numbers that leave no remainder (i.e. Remainder = 0), when it is divided by 9. 900/9 = 100 and remainder is 0. 99/9 = 11 and remainder is 0. Since the numbers are exactly divided by 9, the numbers 81, 18, 900, 99 are multiples of 9.

**Is 72 divisible by 9? ›**

9 is divisible by 9, Therefore, 72 is divisible by 9 and the answer to the question, "Is 72 Divisible By 9?" is Yes.

**Is 40 divisible by 9? ›**

**If the sum of the digits is not divisible by 9, then 40 is not divisible by 9**. 4 is not divisible by 9, Therefore, 40 is not divisible by 9 and the answer to the question, "Is 40 Divisible By 9?" is No.