Divisibility Rule of 3 and 9: Proof and Examples

The divisibility rules of 3 and 9 are very similar. A divisibility rule is a simple and practical method for identifying if a given number is divisible by a fixed divisor without executing the division, often by examining its digits.

• A number is divisible by 3 if the sum of the digits of the given number is multiple of 3 or divisible by 3.
• For example, 85206 is divisible by 3 because the sum of the digits 8 + 5 + 2 + 0 + 6 = 21 is divisible by 3.
• Similarly, a number is divisible by 9 if the sum of the digits of the given number is multiple of 9 or divisible by 9.
• For example, 51993 is divisible by 9 because the sum of the digits 5 + 1 + 9 + 9 + 3 = 27 is divisible by 9.

Key Terms: Divisibility rule, Multiplication and division,  Integers, Addition, Divisibility rule of 3, Divisibility rule of 9, Numbers

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What is the Divisibility Rule of 3?

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According to the divisibility rule of 3, a given number is divisible by 3 if the sum of the digits of the given number is a multiple of 3 or the sum of the digits is completely divisible by 3.

Steps to Find a Given Number Is Divisible by 3 or Not

The following steps can be used to check whether a given number is Divisible by 3 or not.

Let us consider the number is 1434

Step 1

Find the sum of the digits of the given number

1 + 4 + 3 + 4 = 12

Step 2

Check whether the sum is divisible by 3 or a multiple of 3 or not.

• If the sum of the digits is divisible by 3 or a multiple of 3, then the given number is divisible by 3.
• If the sum of the digits is not divisible by 3 or a multiple of 3, then the given number is not divisible by 3.

The sum of the digits is 12 which is a divisible by 3 or a multiple of 3.

Hence the given number 1434 is divisible by 3.

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Proof of Divisibility Rule of 3

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The divisibility rule of 3 can be proved by considering an example.

Let a number be 6435.

The above number can be expanded as

6435 = (6 × 1000) + (4 × 100) + (3 × 10) + (5 × 1)

⇒ 6435 = 6 × (999 + 1) + 4 × (99 + 1) + 3 × (9 + 1) + 5 × 1

⇒ 6435 = (6 × 999 + 4 × 99 + 3 × 9) + (6 × 1 + 4 × 1 + 3 × 1 + 5 × 1)

⇒ 6435 = (6 × 999 + 4 × 99 + 3 × 9) + (6 + 4 + 3 + 5)

We know that the numbers 9, 99, 999,… are divisible by 3, and hence the multiples of these numbers are also divisible by 3.

Therefore the first part (6 × 999 + 4 × 99 + 3 × 9) is divisible by 3.

Now the divisibility of 4368 only depends on the second part (6 + 4 + 3 + 5).

We will observe that the digits 6, 4, 3, and 5 are the digits of the number 6435.

Therefore, we can say that the 6435 is divisible by 3 if the sum of these digits is divisible by 3 or a multiple of 3.

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Examples of Divisibility Rule of 3

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Ques. Is 2367 is divisible by 3?

Ans. The given number is 2367

Step 1: Find the sum of the digits of the given number

2 + 3 + 6 + 7 = 18

Step 2: Check whether the sum is divisible by 3 or a multiple of 3 or not.

The sum of the digits is 18 which is divisible by 3 or a multiple of 3.

Hence the given number 2367 is divisible by 3.

Ques. Is 1562 is divisible by 3?

Ans. The given number is 1562

Step 1: Find the sum of the digits of the given number

1 + 5 + 6 + 2 = 14

Step 2: Check whether the sum is divisible by 3 or a multiple of 3 or not.

The sum of the digits is 14 which is not divisible by 3 or a multiple of 3.

Hence the given number 2367 is not divisible by 3.

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What is the Divisibility Rule of 9?

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According to the divisibility rule of 9, a given number is divisible by 9 if the sum of the digits of the given number is a multiple of 9 or the sum of the digits is completely divisible by 9.

Steps to Find a Given Number Is Divisible by 9 or Not

The following steps can be used to check whether a given number is Divisible by 9 or not.

Let us consider the number is 2925

Step 1

Find the sum of the digits of the given number

2 + 9 + 2 + 5 = 18

Step 2

Check whether the sum is divisible by 9 or a multiple of 9 or not.

• If the sum of the digits is divisible by 9 or a multiple of 9, then the given number is divisible by 9.
• If the sum of the digits is not divisible by 9 or a multiple of 9, then the given number is not divisible by 9.

The sum of the digits is 18 which is a divisible by 9 or a multiple of 9.

Hence the given number 2925 is divisible by 9.

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Proof of Divisibility Rule of 9

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The divisibility rule of 9 can be proved by considering an example.

Let a number be 4275.

The above number can be expanded as

4275 = (4 × 1000) + (2 × 100) + (7 × 10) + (5 × 1)

⇒ 4275 = 4 × (999 + 1) + 2 × (99 + 1) + 7 × (9 + 1) + 5 × 1

⇒ 4275 = (4 × 999 + 2 × 99 + 7 × 9) + (4 × 1 + 2 × 1 + 7 × 1 + 5 × 1)

⇒ 4275 = (4 × 999 + 2 × 99 + 7 × 9) + (4 + 2 + 7 + 5)

We know that the numbers 9, 99, 999,… are divisible by 9, and hence the multiples of these numbers are also divisible by 9.

Therefore the first part (4 × 999 + 2 × 99 + 7 × 9) is divisible by 9.

Now the divisibility of 4275 only depends on the second part (4 + 2 + 7 + 5).

We will observe that the digits 4, 2, 7, and 5 are the digits of the number 4275.

Therefore, we can say that the 4275 is divisible by 9 if the sum of these digits is divisible by 9 or a multiple of 9.

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Examples of Divisibility Rule of 9

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Ques. Is 8721 is divisible by 9?

Ans. The given number is 8721

Step 1: Find the sum of the digits of the given number

8 + 7 + 2 + 1 = 18

Step 2: Check whether the sum is divisible by 9 or a multiple of 9 or not.

The sum of the digits is 18 which is divisible by 9 or a multiple of 9.

Hence the given number 8721 is divisible by 9.

Ques. Is 7823 is divisible by 9?

Ans. The given number is 7823

Step 1: Find the sum of the digits of the given number

7 + 8 + 2 + 3 = 20

Step 2: Check whether the sum is divisible by 9 or a multiple of 9 or not.

The sum of the digits is 20 which is not divisible by 9 or a multiple of 9.

Hence the given number 7823 is not divisible by 9.

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Things to Remember

• A divisibility rule is a simple method for determining whether or not an integer is divisible by a fixed divisor without having to divide it.
• Different integers have different divisibility laws.
• The divisibility rules for 3 and 9 are very similar.
• A given number is divisible by 3 if the sum of the digits of the given number is a multiple of 3 or the sum of the digits is completely divisible by 3.
• A given number is divisible by 9 if the sum of the digits of the given number is a multiple of 9 or the sum of the digits is completely divisible by 9.