Divisibility Rule of 3: Formula, Tricks and Examples

Divisibility Rule of 3 states that a number is divisible by 3 when the sum of its digits is divisible by 3. This page explains the divisibility by 3 rule, digit-sum method, quick checking tricks, solved examples and missing-digit questions used in quantitative aptitude.

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What Is the Divisibility Rule of 3?

A number is divisible by 3 if the sum of its digits is divisible by 3. If the digit sum leaves a remainder of 0 when divided by 3, the original number is also divisible by 3.

For a number such as 4,572, add its digits: 4 + 5 + 7 + 2 = 18. Since 18 is divisible by 3, 4,572 is divisible by 3. The rule works because every power of 10 leaves remainder 1 when divided by 3, so the number and its digit sum have the same remainder upon division by 3.

Divisibility Rule of 3 Formula & Tricks

Important Formulas

Digit-sum test
N is divisible by 3 if S = sum of the digits of N and S mod 3 = 0

Add all digits of N. If the sum is 0, 3, 6, 9, 12, 15, and so on, N is divisible by 3.

Remainder form
N mod 3 = (sum of the digits of N) mod 3

A number and its digit sum leave the same remainder when divided by 3.

Quick Tricks

Reduce the digit sum repeatedly

Add the digits and continue adding the digits of the result until a single digit remains. The final digit is 3, 6 or 9 exactly when the original number is divisible by 3.

Example: For 98,736: 9 + 8 + 7 + 3 + 6 = 33, and 3 + 3 = 6. Therefore, 98,736 is divisible by 3.
Ignore groups whose sum is already a multiple of 3

While adding digits, groups with sums such as 3, 6 or 9 can be removed mentally. The remaining digit sum determines divisibility.

Example: For 7,245, the digits 7 + 2 + 4 + 5 can be grouped as (7 + 2) + (4 + 5) = 9 + 9. Both groups are divisible by 3, so 7,245 is divisible by 3.

Divisibility Rule of 3 Concepts

Applying the Digit-Sum Method

To test a number, add all its digits and check whether the resulting sum is divisible by 3.

If the digit sum is divisible by 3, the original number is divisible by 3. If the digit sum is not divisible by 3, the original number is not divisible by 3. For 8,314, the digit sum is 8 + 3 + 1 + 4 = 16, which is not divisible by 3; therefore, 8,314 is not divisible by 3.

Example: For 56,421, the digit sum is 5 + 6 + 4 + 2 + 1 = 18. Hence, 56,421 is divisible by 3.

Finding a Missing Digit

For a number with a missing digit, choose the digit that makes the total digit sum a multiple of 3.

If the known digits have sum R and the missing digit is x, then R + x must be divisible by 3. The possible digits are 0 through 9, so more than one answer may occur unless an additional condition is given.

Example: In 47x2, the known digit sum is 4 + 7 + 2 = 13. Since 13 + x must be divisible by 3, x can be 2, 5 or 8.

Using the Remainder of the Digit Sum

The remainder of a number upon division by 3 can be found from the remainder of its digit sum.

If the digit sum leaves remainder 1, the number leaves remainder 1; if the digit sum leaves remainder 2, the number leaves remainder 2. For 6,428, the digit sum is 20, and 20 leaves remainder 2 when divided by 3, so 6,428 also leaves remainder 2.

Example: For 73,516, the digit sum is 22. Since 22 = 3 × 7 + 1, the number leaves remainder 1 when divided by 3.

Relation Between Divisibility by 3 and 9

Every number divisible by 9 is also divisible by 3, but a number divisible by 3 need not be divisible by 9.

Use the same digit-sum method for both tests. A digit sum divisible by 3 confirms divisibility by 3, while a digit sum divisible by 9 confirms divisibility by 9. For 126, the digit sum is 9, so 126 is divisible by both 3 and 9. For 123, the digit sum is 6, so it is divisible by 3 but not by 9.

Example: The number 4,572 has digit sum 18. It is divisible by both 3 and 9 because 18 is divisible by both.

Divisibility Rule of 3 Video Lessons

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14 Lessons
Lesson 1 of 14 Quick Revision

Divisibility Rules for 3 and 9

Learn how to test whether a number is divisible by 3 or 9 using the sum of its digits, with clear rules and examples for quantitative aptitude.

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Practice Divisibility Rule of 3 Questions

Practise published questions related to this topic.

1Three numbers are in the ratio of 19 : 7 : 13, and their LCM is 8645. Their HCF is:→ 2When reduced by 6, the smallest number M is divisible by 9, 12, and 26. The digit sum of M is:→ 3If x = √(1671 + √(93 + √49)) , then what is the value of x?→ 4Find the HCF of 44, 110 and 198.→ 5Find the HCF of the numbers 2520, 3600, 5880.→ 6Which of the following number is divisible by both 9 and 11?→ 7If the square root of a number is 0.25 times the number itself, find the number.→ 8What is the smallest 4-digit number that is divisible by 17?→ 9Find the largest number that divides 555, 703, 999 and 1147 leaving the same remainder in each case.→ 10Which number should replace both the '*' in ((*)/21) × ((*)/189) = 1?→

Divisibility Rule of 3 Quick Quiz

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Quick Revision Notes

Divisibility Rule of 3: Quick Revision

Use the digit sum to test divisibility, find remainders and solve missing-digit questions.

  • Add all the digits of the number.
  • The number is divisible by 3 if its digit sum is divisible by 3.
  • The number and its digit sum have the same remainder when divided by 3.
  • For a missing digit x, make the complete digit sum a multiple of 3.
  • A number divisible by 9 is always divisible by 3, but the converse is not always true.
  • Possible one-digit digital roots for multiples of 3 are 3, 6 and 9; zero is also possible for the number 0.

Divisibility Rule of 3 FAQs

What is the formula for the divisibility test for 3?

If S is the sum of the digits of N, then N is divisible by 3 when S mod 3 = 0. Equivalently, N mod 3 = S mod 3.

Is 7,836 divisible by 3?

Yes. Its digit sum is 7 + 8 + 3 + 6 = 24, and 24 is divisible by 3. Therefore, 7,836 is divisible by 3.

What is the smallest digit that can replace x in 52x4 to make it divisible by 3?

The known digit sum is 5 + 2 + 4 = 11. The smallest digit x for which 11 + x is divisible by 3 is 1, giving a total of 12.

Can a number be divisible by 3 but not by 9?

Yes. For example, 123 has digit sum 6, which is divisible by 3 but not by 9. Thus, 123 is divisible by 3 but not by 9.

What is the remainder when 8,245 is divided by 3?

The digit sum is 8 + 2 + 4 + 5 = 19. Since 19 leaves remainder 1 when divided by 3, 8,245 also leaves remainder 1.

Do leading zeroes affect the divisibility test for 3?

No. Leading zeroes add nothing to the digit sum. For example, 0042 has digit sum 6 and is divisible by 3, just like 42.

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