Divisibility Rule of 11

Divisibility Rule of 11 is the rule to check if a number is completely divisible by 11. According to divisibility rule of 11, if the difference between the sum of the digits in a number at odd and even place is zero or divisible by 11, then the number is divisible by 11. It saves time by checking the divisibility of a number by 11 without doing the actual division.

In this article, we will learn what is divisibility rule of 11, how to use he divisibility rule of 11 with examples and solve some questions based on it.

What is Divisibility Rule of 11?

Divisibility Rule of 11 states that if the difference between the sum of digits in a number at odd position and even position is zero or divisible by 11, then the number is divisible by 11.

To Apply the divisibility rule of 11 we follow the below steps:

• Start from the rightmost position, and separate the digits into two groups: digits at odd positions (1st, 3rd, 5th, etc.) and digits at even positions (2nd, 4th, 6th, etc.).
• Subtract the sum of the digits present in the odd group from the sum of the digits present in the even group.
• If the difference is 0 or any value is divisible by 11 with a remainder of 0 then we say the original value is divisible by 11 otherwise it’s not divisible by 11.

Divisibility Rule of 11 with Example

The Divisibility rule for 11 is to calculate the difference between the sum of digits present at an even position to the sum of digits present at an odd position. If this difference is zero or the difference is divisible by 11 then the initial value is divisible by 11.

Example: Check if 70828162317 is divisible by 11

Solution:

To check if 70828162317 is divisible by 11 or not we follow the following steps

Step 1:

Start from the rightmost side, at the digits present at the odd position (1st, 3rd, 5th, and so on ) then add the digit present in the even position (2nd, 4th, 6th, and so on ).

Sum of the Values at Odd Position = 7 + 8 + 8 + 6 + 3 + 7 = 39

Sum of the Values at Even Position = 0 + 2 + 1 + 2 + 1 = 6

Step 2:

Now Calculate the difference between the sum of the values at Even Position and Odd Position

Difference = Sum of the Values at Odd Position – Sum of the Values at Even Position = 39 – 6 = 33

The difference is not equal to zero but the difference 33 is divisible by 11.

Hence the initial value that is 70828162317 is divisible by 11.

Divisibility Rule of 11 for 3 Digit Number

To check if a 3 digit number is divisible by 3 or not, sum the digits at extreme positions and subtract it with middle term. If the difference is zero or multiple of 11, then the number is divisible by 11.

Example 1: Check if 792 is divisible by 11

Solution:

Sum of the first and last digit i.e 7 and 2 is,

7 + 2 = 9

which is equal to the middle digit i.e 9,

Therefore, difference = 9 – 9 = 0

Hence the value is divisible by 11

Example 2: Check if 537 is divisible by 11

Solution:

Sum of the first and last digits i.e 5 and 7 is,

5 + 7 = 12

Difference = 12 – 3 = 9

Since, 9 is not divisible by 11

Hence the value i.e. 537 is NOT divisible by 11

Divisibility Rule of 11 for Large Numbers

Divisibility Rule of 11 is equally valid for larger numbers. Divisibility rule for larger numbers helps in saving time by checking if a number is divisible by 11 or not without performing any actual division. Let’s understand this with an example:

Example 1: Check if 65678932 is divisible by 11 or not.

Solution:

Find the sum of odd digits at odd places = 2 + 9 + 7 + 5 = 23

Find the sum of even digits at odd places = 3 + 8 + 6 + 6 = 23

Difference between sum of digits at odd and even place = 23 – 23 = 0

Since, the Difference between sum of digits at odd and even place is 0. Hence, 65678932 is divisible by 11.

Divisibility Rule of 11 and 12

Divisibility Rule of 11 and 12 are quite different from each other. Divisibility rule of 11 says that the difference between the sum of digits at odd and even places in number is zero or divisible by 11, then the number is divisible by 11. On the other hand, if a number is divisible by 3 and 4 then it is divisible by 12. Let’s see an example.

Example: Check if 132 is divisible by 11 or 12 or both.

Solution:

Checking divisibility by 11

Sum of digits at odd place = 1 + 2 = 3

Sum of digit at even place = 3 as it is the only digit present

Difference = 3 – 3 = 0

Hence, the number 132 is divisible by 11

Checking Divisibility by 12

The number formed by digit at tens and ones place in 132 is 32 which is divisible by 4.

Hence, 132 is divisible by 4

Also, sum of digit of 132 = 1 + 3 + 2 = 6 which is divisible by 3

Hence, 132 is divisible by 3 also.

Since, 132 is divisible by both 4 and 3. Therefore, 132 is divisible by 12.

Therefore, 132 is divisible by both 11 and 12.

Divisibility Rule of 11 Examples

Example 1: Check if the value 102551317418 is divisible by 11.

Solution:

Given value, 102551317418

Difference = Sum of the Values at Odd Position – Sum of the Values at Even Position

Difference = (1 + 2 + 5 + 3 + 7 + 1) – (0 + 5 + 1 + 1 + 4 + 8) = (19) – (19) = 0

The difference is equal to zero hence it is divisible by 11.

Example 2: Check if the value 7954358015 is divisible by 11

Solution:

Given value, 7954358015

Difference = Sum of the Values at Odd Position – Sum of the Values at Even Position

Difference = (7+ 5 + 3 + 8 + 1 ) – (9 + 4 + 5 + 0 + 5 ) = (24) – (23) = 1

The difference is not equal to zero nor 1 is divisible by 11.

Hence the value 7954358015 is not divisible by 11.

Example 3: Find the value X if the value 81X971 is divisible by 11.

Solution:

Given value, 81X971

Since the value is divisible by 11 we can assume that the difference is always equal to 0 or a multiple of 11 represented as 11k.

K is a constant here

Difference = Sum of the Values at Odd Position – Sum of the Values at Even Position = 11K

Difference = (8 + X + 7 ) – (1 + 9 + 1) = 11K

⇒ X + 4 = 11k

As X is a single digit the value needs to be between 0-9 hence we can assume k = 1

X + 4 = 11

X = 7

Hence X = 7 and the value will be 817971

Example 4: Find the value of X if 9X415 is divisible by 11.

Solution:

Given value, 9X415

Since the value is divisible by 11 we can assume that the difference is always equal to 0 or a multiple of 11.

K is a constant here

Difference = Sum of the Values at Odd Position – Sum of the Values at Even Position = 11K

⇒ (9 + 4 + 5 ) – (X + 1) = 11K

⇒ 18 – X -1 =11k

⇒ 17 – X = 11k

As X is a single digit the value needs to be between 0-9 hence we can assume k = 1

⇒ 17 – X = 11

⇒ X = 6

Hence X = 6 and the value will be 96415

Practice Questions on Divisibility Rule of 11

Q1: Check if the value 7825942034 is divisible by 11.

Q2: Find the value X if the value 5471X971 is divisible by 11.

Q3: If the value 5X72Y is divisible by 11 then find the value X-Y?

Q4: Find the smallest 4-digit value divisible by 11.

Divisibility Rule of 11 FAQs

What is the Divisibility Rule of 11?

Divisibility rule for 11 says that if the difference between the sum of digits present at an even position to the sum of digits present at an odd position is zero or divisible by 11 then the number is divisible by 11.

Are there any Exceptions to the Divisibility Rule of 11?

No, there is no exceptions to the rule. It applies to every integer value.

What’s the Difference between the Divisibility rule of 11 and the Divisibility Rule of 9?

The divisibility rule of 9 is to check if the sum of the digits is divisible by 9 whereas the divisibility rule of 11 is a bit different.

Is there any Real Life uses for these Divisibility Rules?

Aside from the fact that it helps in calculation during competitive exams, it is used in various algorithms like for error checking and encryption and decryption processes of cryptography.

What is Divisibility Rule of 12?

A number is divisible by 12 if it is divisible by both 4 and 3.

What is Divisibility Rule of 10?

A number is divisible by 10 if its unit digit is zero

What is Divisibility Rule of 11 for Large Numbers?

Divisibility Rule of 11 for Large Numbers states that if the difference of sum of digits at even place and odd place is zero or multiples of 11 then the number is divisible by 11.