Divisibility Rule of 11: Formula, Test and Examples

Divisibility Rule of 11 checks a number by finding the difference between the sums of alternate digits. If this difference is zero or a multiple of 11, the number is divisible by 11. This page explains the divisibility by 11 rule, formula, calculation method and numerical examples.

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What Is the Divisibility Rule of 11?

A number is divisible by 11 if the difference between the sums of digits in alternate positions is 0 or a multiple of 11.

Starting from the rightmost digit, add digits in alternate positions and separately add the remaining digits. Subtract the smaller sum from the larger sum, or retain the signed difference. If the result is divisible by 11, the original number is divisible by 11. For 121, the alternate sums are 1 + 1 = 2 and 2, so the difference is 0. Therefore, 121 is divisible by 11.

Divisibility Rule of 11 Formula & Tricks

Important Formulas

Alternating Digit Sum
S = (d₀ + d₂ + d₄ + …) − (d₁ + d₃ + d₅ + …)

Here, d₀ is the digit in the units place, d₁ is the tens digit, and so on. The number is divisible by 11 when S is a multiple of 11.

Divisibility Condition
N is divisible by 11 ⇔ S = 0, ±11, ±22, ±33, …

The signed alternating sum may be positive, negative or zero. Any integer multiple of 11 satisfies the test.

Quick Tricks

Add Alternate Digits and Subtract

Mark digits alternately from the right. Add the digits in one group, add the digits in the other group, and find their difference. Check whether the result is 0 or a multiple of 11.

Example: For 2728, the groups are (8 + 7) and (2 + 2). Their difference is 15 − 4 = 11, so 2728 is divisible by 11.
Use the Smaller Difference

The order of subtraction does not matter for the test because a number and its negative are either both multiples of 11 or both not multiples of 11.

Example: For 50644, the alternate sums are 5 + 6 + 4 = 15 and 0 + 4 = 4. The difference is 11, so 50644 is divisible by 11.

Divisibility Rule of 11 Concepts

Applying the Test from the Right

Positions are counted from the units digit, so the rightmost digit belongs to the first alternate group.

For a number written as dₙdₙ₋₁…d₁d₀, add d₀, d₂, d₄ and so on in one group. Add d₁, d₃, d₅ and so on in the other group. This method works for numbers of any length.

Example: For 583924, the groups are (4 + 9 + 5) and (2 + 3 + 8). Their difference is 18 − 13 = 5, so 583924 is not divisible by 11.

Interpreting a Negative Alternating Sum

A negative alternating sum is acceptable when its absolute value is 0 or a multiple of 11.

The divisibility test depends on whether the alternating sum is a multiple of 11, not on its sign. Thus, −11, −22 and −33 satisfy the rule just as 11, 22 and 33 do.

Example: For 918273, the groups are (3 + 2 + 9) and (7 + 8 + 1). The signed difference is 14 − 16 = −2, so the number is not divisible by 11.

Zero Difference Case

If the two alternate-digit sums are equal, the number is divisible by 11.

Equal alternate sums produce an alternating difference of 0, and zero is a multiple of 11. This is the most direct form of the divisibility test.

Example: For 121, the alternate sums are 1 + 1 = 2 and 2. Their difference is 0, so 121 is divisible by 11.

Difference Equal to a Nonzero Multiple of 11

The number is also divisible by 11 when the alternate-digit sums differ by 11, 22, 33 or any other multiple of 11.

The difference does not need to be zero. After calculating the two sums, check whether their difference can be divided exactly by 11.

Example: For 2728, the alternate sums are 8 + 7 = 15 and 2 + 2 = 4. Their difference is 11, so 2728 is divisible by 11.

Divisibility Rule of 11 Video Lessons

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Divisibility Rule for 11

Learn the divisibility rule for 11 and understand how to apply it to determine whether a given number is divisible by 11.

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Practice Divisibility Rule of 11 Questions

Practise published questions related to this topic.

1The LCM of the numbers 7, 12.4 and 1 is:→ 2If 15² = 225 , then find the approximate value of √22.5+√2.25+25.→ 3A = 25 x 32, B = 28 x 34. What is the product of Highest Common Factor and Least Common Multiple of A and B?→ 473P2 is divisible by 8. Which of the following cannot be the value of P?→ 5Which of the following number is neither divisible by 3 nor by 4?→ 6Find the smallest positive integer that leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, and a remainder of 4 when divided by 5.→ 7Find the smallest number which when divided by 9, 12 and 15 leaves a remainder 3 in each case.→ 8Find the square root of 0.000324.→ 9Two buses leave a station at the same time. Bus A returns every 48 minutes and Bus B returns every 72 minutes. After how many hours will both buses return together at the station for the first time?→ 10Three bells ring at intervals of 18 seconds, 24 seconds, and 30 seconds. If they all start ringing together at 10:00:00 AM, after how many minutes will they ring together again?→

Divisibility Rule of 11 Quick Quiz

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Quick Revision Notes

Divisibility Rule of 11: Quick Revision

Use the alternate-digit sum method to test divisibility by 11.

  • Count alternate positions from the units digit.
  • Add the digits in each alternate group separately.
  • Find the signed or absolute difference between the two sums.
  • The number is divisible by 11 if the difference is 0 or any multiple of 11.
  • The sign of the difference does not affect the result.
  • For 12345, (5 + 3 + 1) − (4 + 2) = 3, so it is not divisible by 11.

Divisibility Rule of 11 FAQs

What is the formula for the divisibility test for 11?

For N = dₙdₙ₋₁…d₁d₀, calculate S = (d₀ + d₂ + d₄ + …) − (d₁ + d₃ + d₅ + …). N is divisible by 11 if S is a multiple of 11.

Is a number divisible by 11 when the alternating difference is 0?

Yes. Zero is a multiple of 11, so an alternating difference of 0 means the number is divisible by 11. For example, 121 gives (1 + 1) − 2 = 0.

Is 2728 divisible by 11?

Yes. The alternate sums are 8 + 7 = 15 and 2 + 2 = 4. Their difference is 11, which is divisible by 11.

Is 12345 divisible by 11?

No. Its alternate sums are 5 + 3 + 1 = 9 and 4 + 2 = 6. The difference is 3, not a multiple of 11.

Does the order of subtraction matter in the divisibility rule of 11?

No. The differences 11 and −11 both represent multiples of 11. Therefore, either alternate sum can be subtracted from the other.

How do you test 50644 for divisibility by 11?

The alternate sums are 5 + 6 + 4 = 15 and 0 + 4 = 4. Since 15 − 4 = 11, 50644 is divisible by 11.

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