Divisibility Rule of 9: Rule, Formula and Examples

Divisibility Rule of 9 states that a number is divisible by 9 when the sum of its digits is divisible by 9. This page explains the divisibility by 9 rule, digit-sum method, useful shortcuts, unknown-digit questions and solved numerical examples.

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What is the Divisibility Rule of 9?

A number is divisible by 9 if the sum of its digits is divisible by 9. The sum may be exactly 9, 18, 27, or any other multiple of 9.

For a number N = 100a + 10b + c, we have 100 ≡ 1, 10 ≡ 1 and 1 ≡ 1 (mod 9). Therefore, N and a + b + c leave the same remainder when divided by 9. For example, the digit sum of 72,936 is 7 + 2 + 9 + 3 + 6 = 27, and 27 is divisible by 9. Hence, 72,936 is divisible by 9.

Divisibility Rule of 9 Formula & Tricks

Important Formulas

Divisibility test
N is divisible by 9 ⇔ sum of the digits of N is divisible by 9

Add all the digits of the number. If the resulting sum is a multiple of 9, the original number is divisible by 9.

Remainder rule
N mod 9 = (sum of the digits of N) mod 9

A number and its digit sum have the same remainder when divided by 9.

Unknown digit condition
Known digit sum + x ≡ 0 (mod 9)

For a missing digit x, choose a digit from 0 to 9 that makes the total digit sum a multiple of 9.

Quick Tricks

Reduce the digit sum repeatedly

If the digit sum is large, add its digits again until a single digit or a clear multiple of 9 is obtained. A final digit of 9, or a digit sum that reduces to 9, indicates divisibility by 9.

Example: For 8,764,539, the digit sum is 42. Since 4 + 2 = 6, the number is not divisible by 9.
Find a missing digit by the next multiple of 9

Add the known digits and determine the digit needed to reach the next multiple of 9. If the known sum is already a multiple of 9, the missing digit can be 0 or 9, depending on the required position and conditions.

Example: In 43x72, the known sum is 4 + 3 + 7 + 2 = 16. The next multiple of 9 is 18, so x = 2.
Check only the digit sum

The place values do not need to be calculated. Addition, subtraction or rearrangement of digits does not affect the divisibility test when the resulting digit sum is checked correctly.

Example: Both 1,236 and 6,321 have digit sum 12, so neither is divisible by 9.

Divisibility Rule of 9 Concepts

Applying the digit-sum test

Add every digit of the number and check whether the sum is a multiple of 9.

The multiples of 9 include 0, 9, 18, 27, 36, 45 and so on. If the digit sum is one of these values, the number passes the divisibility test for 9.

Example: For 54,729, the digit sum is 5 + 4 + 7 + 2 + 9 = 27. Since 27 = 9 × 3, 54,729 is divisible by 9.

Using repeated digit sums

A repeated digit sum can be used to test divisibility when the first sum is large.

Continue adding the digits of the sum until a single digit is obtained. A final value of 9 means divisibility by 9; any final value from 1 to 8 means the number is not divisible by 9. A digit sum of 0 occurs for zero and is also divisible by 9.

Example: For 98,765,432, the digit sum is 44, and 4 + 4 = 8. Therefore, the number is not divisible by 9.

Finding an unknown digit

For a number containing an unknown digit, make the total digit sum a multiple of 9.

First add the known digits. If their sum is S, the missing digit x must satisfy S + x = 9k for some integer k, with 0 ≤ x ≤ 9. The answer may require checking the allowed digit range or any condition such as a non-zero first digit.

Example: For 7x438, the known digit sum is 7 + 4 + 3 + 8 = 22. The smallest digit that makes a multiple of 9 is x = 5, because 22 + 5 = 27.

Divisibility and remainders

A number has the same remainder as its digit sum when divided by 9.

Since 10 ≡ 1 (mod 9), every place value 10^n is also congruent to 1 modulo 9. Thus, the remainder of a number can be found by taking the remainder of its digit sum. For instance, a digit sum of 23 gives remainder 5 because 23 = 9 × 2 + 5.

Example: The digit sum of 6,482 is 20. Therefore, 6,482 leaves remainder 2 when divided by 9.

Divisibility Rule of 9 Video Lessons

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Lesson 1 of 14 Quick Revision

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Practice Divisibility Rule of 9 Questions

Practise published questions related to this topic.

1Which of the following number is NOT divisible by 11 ?→ 2Find the HCF of 2.4, 1.2 and 0.6.→ 3Three tankers contain 1134 litres, 1215 litres and 513 litres of milk, respectively. Find the capacity of a container (in litres) which can empty the milk of the 3 tankers an exact number of times.→ 4The LCM of the numbers 5.4 and 0.384 is:→ 5What is the least number that should be added to 517 so that the number obtained is exactly divisible by 15, 25 and 8?→ 6Least Common Multiple of A and B is 18. Least Common Multiple of C and D is 24. What is the Least Common Multiple of A, B, C and D?→ 7What is the Highest Common Factor of 15, 25 and 35?→ 8What is the largest 2 digit number which is exactly divisible by both 2 and 7?→ 9A record entry gives √32 . Simplify.→ 10If x = √81 ÷ √9+√36, then what is the value of 2x?→

Divisibility Rule of 9 Quick Quiz

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Quick Revision Notes

Divisibility Rule of 9: Quick Revision

Use the digit-sum property to test divisibility, find missing digits and calculate remainders.

  • A number is divisible by 9 if and only if its digit sum is divisible by 9.
  • Relevant digit sums are multiples of 9: 0, 9, 18, 27, 36 and so on.
  • A number and its digit sum have the same remainder when divided by 9.
  • For an unknown digit x, use known digit sum + x = a multiple of 9.
  • Repeatedly adding the digits is valid for checking divisibility by 9.
  • The divisibility test depends on the digits, not on their order.

Divisibility Rule of 9 FAQs

What is the divisibility by 9 rule?

A number is divisible by 9 when the sum of all its digits is divisible by 9. For example, 4,572 has digit sum 18, so it is divisible by 9.

Is 0 divisible by 9?

Yes. Since 0 = 9 × 0, zero is divisible by 9.

How do you check whether 7,836 is divisible by 9?

Add its digits: 7 + 8 + 3 + 6 = 24. Since 24 is not divisible by 9, 7,836 is not divisible by 9.

What is the remainder when 58,247 is divided by 9?

Its digit sum is 5 + 8 + 2 + 4 + 7 = 26. Since 26 leaves remainder 8 on division by 9, 58,247 also leaves remainder 8.

Which digit should replace x in 62x14 so that the number is divisible by 9?

The known digit sum is 6 + 2 + 1 + 4 = 13. The next multiple of 9 is 18, so x = 5.

Does changing the order of digits affect divisibility by 9?

No. Rearranging digits does not change their sum. Therefore, 126 and 612 have the same divisibility result for 9.

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