# Divisibility Rule of 11

Divisibility rule of 11 or any divisibility rule for that matter is a shortcut used to find if a number is perfectly divisible by that number or not. These rules are usually a set of conditions which need to be met by the number or a set of operations or modifications which need to be performed on the number to verify its divisibility.

Let us look into the divisibility rule of 11

Index

## Divisibility rules of 11

Interestingly, there are two divisibility rules for 11.

### Rule 1

This rule is different for numbers with an even number of digits and numbers with an odd number of digits.

Numbers with an odd number of digits:

Steps involved in the process of finding the divisibility of a number with an odd number of digits by 11:

1: Separate the first and last digits of the given number.
2: Subtract the remaining number from step 1, by the removed digits.
3: Repeat steps 1 and 2 until they cannot be further performed.
4: If the resulting number at the end of this process is divisible by 11, then the original number is also divisible by 11.

Numbers with an even number of digits:

Steps involved in the process of finding the divisibility of a number with an even number of digits by 11:

1: Separate the first and last digits of the given number.
2: Subtract the last digit and add the first digit of the original number from the number obtained in step 1.
3: Repeat steps 1 and 2 until they cannot be further performed.
4: If the resulting number at the end of step 3 is divisible by 11, then the original number is also divisible by 11.

#### Examples

Q. Verify if the number 12321 is divisible by 11 or not.

AnsStep 1: Separating the first and last digits of the given number.

12321 = 232 + 1 + 1

Step 2: Subtracting the number obtained in step 1, by the removed numbers.

232 – 1 – 1 = 230

Step 3: Repeating steps 1 and 2 until they cannot be further performed.

230 = 3 + 2 + 0

3 – 2 – 0 = 1

Step 4: If the resulting number after step 3 is divisible by 11, then the original number is also divisible by 11.
1 is not divisible by 11, so 12321 is also not divisible by 11.

Q. Verify if the number 918082 is divisible by 11 or not.

Ans. Step 1: Separating the first and last digits of the given number.

918082 = 1808 + 9 + 2

Step 2: Subtracting the last digit and adding the first digit to the number obtained in step 1.

1808 – 2 + 9 = 1815

Step 3: Repeating steps 1 and 2 until they cannot be further performed.

1815 = 81 + 1 + 5

81 – 5 + 1 = 77

Step 4: If the resulting number after step 3 is divisible by 11, then the original number is also divisible by 11.

77 is divisible by 11, so 918082 is also divisible by 11.

### Rule 2

Steps involved in finding if a number is divisible by 11:

1: Label each digit of the given number as odd and even alternatively starting from the right.
2: Add the digits with the same labels.
3: Subtract the sums obtained in step 2 and take their modulus.
4: If the final number obtained in step 3 is either divisible by 11 or 0, then the original number is divisible by 11.

#### Examples

Q. Verify if the number 918082 is divisible by 11 or not.

Ans. Step 1: Labeling each digit as odd and even alternatively starting from the rightmost digit.

9 1 8 0 8 2

o e o e o e

Step 2: Adding the digits with the same labels.

o: 9 + 8 + 8 = 25

e: 1 + 0 + 2 = 3

Step 3: Subtracting the sums obtained in step 2 and taking their modulus.

|o – e| = |25 – 3| = 22

Step 4: If the number obtained in step 3 is either divisible by 11 or 0, then the original number is divisible by 11.

22 is divisible by 11, so the given number 918082 is also divisible by 11.

## FAQs

What is the divisibility rule of 11?

There are 2 divisibility rules for 11.

The first rule states that, first the first and last digits of the given number must be removed, then if the number has

Even number of digits then the first digit must be added and the last digit must be subtracted from it, whereas, if the given number has an odd number of digits, then both the first and last digits must be subtracted from the remaining number, if the number obtained at the end of this process is divisible by 11(including 0), then the original number is also divisible by 11.

The second rule states that, the digits of the given number must be labeled odd and even starting from the rightmost digit, add those digits with the same label and then subtract the obtained sums and find its modulus. If the number obtained at the end of this process is divisible by 11(including 0), then the original number is also divisible by 11.

What are divisibility rules?

Divisibility rules can be considered as shortcuts which can be used to find if a given number is divisible by a particular divisor or not.

Where were divisibility rules first introduced and by whom?

Divisibility rules were first introduced in the mathematical games column of the popular science magazine, Scientific American, written by Martin Gardner, an American popular science and popular mathematics writer.

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