Divisibility Rule of 11
The divisibility rule of 11 is a simple mental calculation that checks if the number 11 completely divides another number. The divisibility by 11 rule states that if the difference between the sum of the digits at the odd and even places equals 0 or divisible by 11, then the number is divisible by 11.
What is the Divisibility Rule of 11?
The divisibility by 11 rule states that if the difference of the sum of the digits at odd places and even places of the number is 0 or divisible by 11, then the given number is also divisible by 11. For calculating the difference, we always subtract the smaller value from the larger value. The divisibility rule of 11 helps us to know whether a given number is completely divisible by 11, without leaving any remainder, or not. Divisibility rules are a set of rules or condition that checks if an integer is completely divisible by another number without any remainder and without actually performing the division. It is a known fact that 11, 22, 33, 44.... are multiples of 11, that occur in the multiplication table. So, we can clearly say that these numbers are divisible by 11. For example, 33/11 = 3 and the remainder is 0. It is easy to do the divisibility test of 11 for smaller numbers. But there are situations wherein we will have to check if a large number is divisible by 11 or not. In such cases, we use the divisibility rule of 11 to draw a conclusion.
Look at the image given below showing steps of divisibility by 11. But, note that it is not mandatory to start from the leftmost digit to check the divisibility by 11, we can even start from the rightmost digit.
Divisibility Rule of 11 For Large Numbers
As we know from the divisibility rule of 11, a number is divisible by 11 if the difference between the sum of the digits at the odd and the even places are either equal to 0 or that can be completely divided by 11 without leaving a remainder. But how do we find if a larger number is divisible by 11? For example, how do we find if the number 2541 is divisible by 11 or not? To check this, let us apply the divisibility test by 11 to the number 2541. In the number 2541, the digits at the odd positions are 2 and 4 (if we start from the left), hence the sum is 6. The numbers at the even positions are 5 and 1, hence their sum is 6. Now, the difference between the sums obtained is 6  6, which is equal to 0. We know that 0 is divisible by every number, so it is divisible by 11. Therefore, the number 2541 is divisible by 11.
Divisibility Rule of 11 and 12
Divisibility Rules of 11 and 12 are different. In the divisibility rule of 11, we check to see if the difference between the sum of the digits at the odd places and the sum of the digits at even places is equal to 0 or a number that is divisible by 11, whereas the divisibility rule of 12 states that a number is divisible by 12 if it is completely divisible by both 3 and 4 without leaving a remainder. Now, let us take a number and check for the divisibility rule of 11 and 12.
Number = 764852
Let us check if divisibility by 11 is true.
Sum of the digits at odd places (from the left) = 7 + 4 + 5 = 16
Sum of the digits at even places = 6 + 8 + 2 = 16
Difference between the sum of the digits at odd and even places = 16  16, which is 0.
Therefore, 764852 is divisible by 11.
Let us check if it is divisible by 12 or not.
For this let us check if the number is divisible by both 3 and 4. Sum of all the digits = 7+6+4+8+5+2 = 32. The Sum of 3 and 2 is 5. 5 can not be divided by 3 completely. Therefore, 764852 is not divisible by 3. Let us also check the divisibility by 4. For a number to be divisible by 4, the last two digits of the number should be either '00' or a number divisible by 4. In the given number, the last two digits are 52. When 52 is divided by 4, the quotient is 13 and the remainder is 0. Hence, we can say that the number 764852 is divisible by 4.
But for a number to be divisible by 12, it should pass the divisibility test of 3 as well as 4. Here, we see that the number is not divisible by 3. So we can say that it is not divisible by 12. From the example, we can understand that divisibility rules for 11 and 12 are totally different and it is not necessary that a number that is divisible by 11 should be divisible by 12 also.
Divisibility Test of 11 and 7
Divisibility Rules of 11 and 7 are also totally different. Let us have a brief note on the rules and then work on an example. To check if a number is divisible by 7, pick the last digit of the number (unit place digit), double it, and then subtract it from the rest of the number. If the difference obtained is divisible by 7, then we can say that the number is divisible by 7. If we are not sure about the difference being a multiple of 7, then repeat the same process with the rest of the digits. The divisibility rule of 11 is already discussed. Now let is work out the divisibility rule of 11 and 7 for the same number.
Let us check if the number 16387 is divisible by 11 and 7.
Firstly let us check if 16387 is divisible by 11. To know that let us find out the difference between the sum of the digits at the odd and even places.
Sum of digits at odd places (from the left) = 1+3+7 = 11
Sum of digits at even places = 6 + 8 = 14
Difference between the digits at odd and even places = 14  11 = 3
3 is not divisible by 11. Therefore, 16387 is not divisible by 11.
Let us now check if 16387 is divisible by 7.
The last digit is 7. Multiplying the last digit by 2, we get 7 × 2 = 14. Now we subtract 14 with the rest of the digits in the number which is 1638. Therefore, 1638  14 = 1624. We repeat the same process for 1624. The last digit of 1624 is 4. Multiplying by 2, we get 4 × 2 = 8. Subtracting it from the rest of the digits, which is 162, we get, 162  8, which is 154. Let us repeat the same step for 154. Multiplying the last digit by 2, we get 4 × 2 which is 8. Subtracting it from the rest of the digits which is 15, we get 15  8, which is 7. 7 is a multiple of 7 and it divides completely without leaving a remainder. Therefore, we can say that 16387 is divisible by 7.
So, we can conclude that 16387 is divisible by 7 and not divisible by 11.
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Divisibility Rule of 11 Examples

Example 1: Using the divisibility rule of 11, find out if the following numbers are divisible by 11 or not.
a) 4563
b) 981375
Solution:To check if a number is divisible by 11 or not, we find the difference between the sum of the digits at the odd and even positions starting from the leftmost digit or the rightmost digit of the number. If the difference is either 0 or a number that 11 completely divides without leaving a remainder, then it is divisible by 11.
a) 4563
Sum of the digits at odd places (from the left) = 4 + 6 = 10
Sum of the digits at even places = 5 + 3 = 8
Difference = 10  8 = 2.
Therefore, 2 is not divisible by 11. So, 4563 is not divisible by 11.
b) 981375
Sum of the digits at odd places (from the left) = 9 + 1 + 7 = 17
Sum of the digits at even places = 8 + 3 + 5 = 16
Difference = 17  16 = 1.
Therefore, 1 is not divisible by 11. So, 981375 is not divisible by 11. 
Example 2: Using the divisibility rule of 11, find which of the following 5digit numbers is divisible by 11.
a) 10000 b) 10010 c) 10020
Solution:
Let us check all the three numbers for divisibility by 11.a) 10000
Sum of digits at odd places = 1 + 0 + 0 = 1
Sum of digits at even places = 0 + 0 = 0
Difference = 1  0 = 11 is not divisible by 11. Therefore, 10000 is not divisible by 11.
b) 10010
Sum of digits at odd places = 1 + 0 + 0 = 1
Sum of digits at even places = 0 + 1 = 1
Difference = 1  1 = 00 divided by 11 gives 0 as the remainder. Therefore, 10010 is divisible by 11.
b) 10020
Sum of digits at odd places = 1 + 0 + 0 = 1
Sum of digits at even places = 0 + 2 = 2
Difference = 2  1 = 11 is not divisible by 11. Therefore, 10020 is not divisible by 11.

Example 3: Are all numbers that end with 11, divisible by 11? Explain with an example.
Solution:
No, not all the numbers that end with 11, are divisible by 11. For example, let us take the number 1011. The last two digits are 11 here. Applying the divisibly rule of 11 for 1011,
Sum of digits at odd places (from the left) = 1 + 1 = 2
Sum of digits at even places = 0 + 1 = 1
Difference = 2  1 = 1
1 is not divisible by 11, so 1011 is not divisible by 11. Therefore, not all the digits that end with 11 are divisible by 11.
Divisibility Rule of 11 FAQs
What is the Divisibility Rule of 11 in Math?
A number is said to be divisible by 11 if the difference between the sum of the digits at the alternative places starting from the left or the right is either 0 or a number that is divisible by 11. For example, in the number 7480, the sum of digits at the odd position is 7 + 8, which is 15 and the sum of digits at the even position is 4 + 0, which is 4. The difference between 15 and 4 is 11. 11 can be completely divided by 11 with 0 as the remainder. Therefore, 7480 is divisible by 11.
How do you Check if Large Numbers are Divisible by 11?
Numbers like 11, 22, 33, 44, and so on are easy to check if they are divisible by 11, as they appear in the multiplication table of 11, which is easy to remember. To check if a larger number is divisible by 11, find the difference between the sum of the digits from the odd places and the sum of the digits at the even places and check if it is 0 or multiples of 11. If yes, then the number is divisible by 11. For example, in the number 111111, the sum of digits at odd places starting from the left is 1 + 1 + 1 = 3 and the sum of digits at even places starting from the left is 1 + 1 + 1 = 3. Therefore, the difference is 3  3, which is 0. Therefore, 111111 is divisible by 11.
Are All Numbers Divisible by 11 are Also Divisible by 7?
No, not all the numbers that are divisible by 11 are divisible by 7. For example, the number 121 is divisible by 11, since the difference between the digits at the odd and the even places are 0, whereas it is not divisible by 7. To check that, let us double the unit place digit, which is 1. So 1 x 2 is 2, Now, we subtract it from the rest of the number which is 12. So 12  2 = 10. 10 is not a multiple of 7. Hence it is not divisible by 7. Therefore, not all the numbers that are divisible by 11 are divisible by 7.
What is the Smallest and the Largest 4Digit Number Divisible by 11?
1001 is the smallest and 9999 is the largest 4 digit number divisible by 11. In 1001, the difference between the sum of the digits at the odd and even places starting from the left to right is (1+0)  (0+1), which is 0. Similarly, for 9999, the difference between the sum of the digits at the odd and even places starting from the left to right is (9 + 9)  (9 +9), which is 18 18 or 0. Therefore, 1001 and 9999 are divisible by 11.
Using the Divisibility Rule of 11, check if 1111111 is Divisible By 11?
No, 1111111 is not divisible by 11. This is because the difference between the sum of the digits at the odd and the even places starting from the leftmost digit is not 0 or a number that is divisible by 11. Sum of the digits at the odd places = 1 +1+1+1 =4, Sum of the digits at the even places = 1 + 1 + 1 = 3. Therefore, the difference is 4  3, which is 1. Since 1 is not divisible by 11, we can say that 1111111 is not divisible by 11.
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