Vedic Maths – Special Division Methods : Divisibility By 11

The divisibility rule is a basic concept in mathematics that would determine if a given number is divided by a particular divisor or not without actually performing the entire division process. Let us understand the divisibility rule of 11 and some easy tricks to solve problems related to 11 in this article.

Divisibility rule of 11

Let us discuss a few methods through which, we can say if a number is divisible by 11.

Method 1: If the difference between the sum of the digits in odd places and the sum of the digits in even places is equal to zero(0) or 11, then the given original number is divisible by 11.

let us understand this through an example,

Q 1. Check if 71049 is divisible by 11 or not.

Solution: Given number = 71049

sum of digits at odd places = 9+0+7 = 16

sum of digits at even places = 4+1 = 5

their difference = 16 – 5 = 11

Hence, given number 71049 is divisible by 11.

Method 2 : Follow the below mentioned steps for this method.

Step 1: Let us consider the pattern, -1+1-1+1-1+1-1+1 . . . . . . .

Step 2: write this pattern , below the given number

let us consider the given number as 725307

so, 7 2 5 3 0 7

-1 +1 -1 +1 -1 +1

Step 3: now, as shown in the figure below, multiply each digit in the number with its corresponding digit in the pattern.

Divisibility by 11

Step 4: Add all the digits that we get in the previous step.

-7+2-5+3+0+7 = 0

If the result, that we get in this step is equal to 0 or 11 or a multiple of 11 , then the original given number is divisible by 11. since, we got 0 in this step, the given number 725307 is divisible by 11. Let us also look at a few easy techniques in multiplication and division with 11.

Process for easy division with 11 :

let us consider an example, 12345 ÷ 11

Step 1 : write the first digit of the given number as it is.

Step 2: subtract the next digit of the number with the digit we got in the previous step.

Divisibility by 11

Step 3: The last number that we got in the above step is our remainder. Hence, on dividing 12345 by 11, we have quotient =1122 and remainder = 3

Process for easy Multiplication with 11 :

Let us consider an example, 18 × 11

Step 1: Write the first and last digit of the given number as it is.

Step 2: write the sum of the digits of the given number in between the digits of step 1.

1 (1+8) 8

Hence, 198 is the required result of the given multiplication 18 × 11

Let us make our concepts more clear with more examples.

Q 2. Check if 105875 is divisible by 11 or not?

Solution: Sum of digits at odd places = 5+8+0 = 13

Sum of digits at even places = 7+5+1 = 13

Difference = 13 – 13 => 0

Hence, 105875 is divisible by 11.

Q 3. When 121011 is divided by 11, the remainder we get is ?

Solution : Let us follow the simple trick,

List of divisibility by 11

Hence remainder = 0

Q 4. Check if the number 1001011 is divisible by 11 or not?

Solution: on following the method2 as said above, we have

Divisibility by 11

sum = -1 + 0+ 0+ 1+ 0 +1 -1 => 0 Hence, given number is divisible by 11.

Q 5. The smallest number which should be added to 3428 so as to obtain a multiple of 11, is?

Solution : Let us first divide the given number by 11.

3428 ÷ 11

Divisibility by 11

remainder = 7

If remainder becomes 11, then the number would have exactly divisible by 11. So, the remainder, in order to become 11, we must add 4 to it. So, here the least number to be added to make the number divisible by 11 is 4.

Q 6. Find the value of x , in 4321×8 in order to make it completely divisible by 11?

Solution : Sum of digits at odd places = 8+1+3 = 12

sum of digits at even places = x+2+4 = 6+x

now, 12-(6+x)=0 or 12 -(6+x)=11

on solving both, 12-6-x=0 or 12-6-x = 11

x=6 or x=-5(Impossible)

so, value of x = 6