Divisibility Rule of 18: Formula, Trick and Examples

The Divisibility Rule of 18 states that a number is divisible by 18 if it is divisible by both 2 and 9. Therefore, its last digit must be even and the sum of its digits must be divisible by 9. This page explains the rule, shortcut method and solved numerical examples.

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What is the Divisibility Rule of 18?

A number is divisible by 18 if its last digit is even and the sum of its digits is divisible by 9.

Since 18 = 2 × 9 and 2 and 9 are co-prime, divisibility by both 2 and 9 guarantees divisibility by 18. For example, 1,152 ends in 2, and its digit sum is 1 + 1 + 5 + 2 = 9. Hence, 1,152 is divisible by 18.

Divisibility Rule of 18 Formula & Tricks

Important Formulas

Divisibility condition
N ÷ 18 is an integer ⇔ N is divisible by 2 and 9

Check the last digit for divisibility by 2 and the digit sum for divisibility by 9.

Test for 2
Last digit ∈ {0, 2, 4, 6, 8}

A number is divisible by 2 when its units digit is even.

Test for 9
Sum of digits ≡ 0 (mod 9)

A number is divisible by 9 when its digit sum is a multiple of 9.

Quick Tricks

Use the two-part check

Check the last digit first. If it is odd, the number cannot be divisible by 18. If it is even, add the digits and check whether the sum is 9, 18, 27, 36, or another multiple of 9.

Example: For 4,698, the last digit 8 is even and 4 + 6 + 9 + 8 = 27. Therefore, 4,698 is divisible by 18.
Reject quickly using either condition

Both conditions are compulsory. An even number with a digit sum not divisible by 9, or a number with a digit sum divisible by 9 but an odd last digit, is not divisible by 18.

Example: 738 has digit sum 18, but its last digit is 8, so it is divisible by 18. In contrast, 729 has digit sum 18 but ends in 9, so it is not divisible by 18.

Divisibility Rule of 18 Concepts

Combined test for divisibility by 2 and 9

To test divisibility by 18, apply the divisibility tests for 2 and 9 together.

The last digit must be one of 0, 2, 4, 6 or 8. Also, the digit sum must be a multiple of 9. If either condition fails, the number is not divisible by 18.

Example: 2,934 ends in 4 and has digit sum 2 + 9 + 3 + 4 = 18. Thus, 2,934 is divisible by 18.

Checking a number that fails the even-digit condition

A number with an odd last digit cannot be divisible by 18, even if its digit sum is divisible by 9.

Divisibility by 18 requires divisibility by 2. Numbers ending in 1, 3, 5, 7 or 9 are not divisible by 2 and therefore cannot be divisible by 18.

Example: 1,269 has digit sum 1 + 2 + 6 + 9 = 18, but it ends in 9. Hence, it is not divisible by 18.

Checking a number that fails the digit-sum condition

An even last digit alone does not prove divisibility by 18; the digit sum must also be divisible by 9.

If the units digit is even but the digit sum is not a multiple of 9, the number is divisible by 2 but not by 18.

Example: 2,346 ends in 6, but its digit sum is 2 + 3 + 4 + 6 = 15. Since 15 is not divisible by 9, 2,346 is not divisible by 18.

Finding a missing digit for divisibility by 18

For a number with a missing digit to be divisible by 18, the completed number must end in an even digit and have a digit sum divisible by 9.

First apply the units-digit condition if the missing digit is in the units place. Then use the digit-sum condition. A digit can be selected from 0 to 9 only if it satisfies both conditions.

Example: For 5□4, the last digit is already even. The known digit sum is 5 + 4 = 9, so the missing digit must be 0 or 9 to make the total a multiple of 9. Both 504 and 594 are divisible by 18.

Divisibility Rule of 18 Video Lessons

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Divisibility Rules for 6, 12, 15 and 18

Learn the divisibility rules for 6, 12, 15, and 18, and understand how to apply these tests to determine whether numbers are divisible by each divisor.

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Practice Divisibility Rule of 18 Questions

Practise published questions related to this topic.

1The product of two numbers is 188 and their LCM is 94. Find their HCF.→ 2Find the least number which leaves remainder 3 when divided by 5, 7, and 8.→ 3The square root of the sum of squares of two numbers is 25. If one number is 15, find the other number.→ 4The LCM of the numbers 11.4 and 0.42 is:→ 5What is the smallest number which when increased by 30 is exactly divisible by 6, 10 and 12?→ 6What is the least number that should be added to 443 so that the number obtained is exactly divisible by 12, 5 and 6?→ 7Least Common Multiple of A and B is 36 and C = 24. What is the Least Common Multiple of A, B and C?→ 8A positive number exceeds its square root by 132. Find the number given that it is also a multiple of 12.→ 9The HCF of two numbers is 24, their LCM is 360, and their difference is 48. Find the numbers.→ 10Find the least number that must be added to 5600 to make it a perfect square.→

Divisibility Rule of 18 Quick Quiz

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Quick Revision Notes

Quick Revision Points

Use these rules to test divisibility by 18.

  • 18 = 2 × 9, and 2 and 9 are co-prime.
  • The last digit must be 0, 2, 4, 6 or 8.
  • The sum of the digits must be divisible by 9.
  • Both conditions must be satisfied; either condition alone is insufficient.
  • A number divisible by 18 is also divisible by 2, 9, 6 and 3.
  • For a missing-digit question, check both the units digit and the digit sum.

Divisibility Rule of 18 FAQs

What is the divisibility test for 18?

A number is divisible by 18 if its last digit is even and the sum of its digits is divisible by 9.

Is every even number with a digit sum divisible by 9 divisible by 18?

Yes. An even number is divisible by 2, and a digit sum divisible by 9 means the number is divisible by 9. Therefore, it is divisible by 18.

Is 5,832 divisible by 18?

Yes. It ends in 2, which is even, and its digit sum is 5 + 8 + 3 + 2 = 18, a multiple of 9.

Why is 729 not divisible by 18?

Its digit sum is 7 + 2 + 9 = 18, so it is divisible by 9, but its last digit is odd. It is not divisible by 2 and therefore not by 18.

Is 0 divisible by 18?

Yes. 0 ÷ 18 = 0, which is an integer. Its last digit is even and its digit sum is 0, a multiple of 9.

How can the missing digit in 4□6 be found if the number must be divisible by 18?

The last digit 6 already satisfies the divisibility-by-2 condition. The digit sum is 4 + □ + 6 = 10 + □, so the missing digit must be 8 to make the sum 18. The number is 486.

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