Divisibility Rule of 11

The divisible rule is a sort of shortcut in maths that helps the candidates to identify the number whether divisible by another number. “The divisibility rule of 11” is a topic of the GMAT exam. The divisibility rule of 11 determines whether 11 totally divides another integer. According to the rule for the divisibility of 11, an integer is divisible by 11 if the difference between the sums of its alternative digits is either 0 or divisible by 11. This topic is a type of GMAT Problem Solving Question. This problem solving question deals with the quantitative problems that test the numerical literacy of the candidates.

What Is the Divisibility Rule of 11? 

The divisibility rule of 11 implies the following:
If the difference between the sum of numbers at even and odd places is zero or divisible by 11, then the provided number will be divisible by 11.
The smaller value needs to be constantly deducted from the larger value to determine the difference. The rule of divisibility of 11 enables the candidates to determine whether a given integer is entirely divisible by 11 without any remainders. Divisibility rules are a collection of criteria that, without actually doing the division, determine whether an integer is entirely divisible by another number.
The multiples of 11 that appear in the multiplication table include 11, 22, 33, 44, and so on. Therefore, it is evident that these numbers can be divided by 11. For instance, 33/11 = 3, leaving 0 as the remainder. For lower integers, it is simple to do the divisibility test of 11. There are, however, circumstances in which we must determine whether a huge number is divisible by 11 or not. In these circumstances, we get a conclusion using the divisibility rule of 11.
See the illustration below for the progression of divisibility by 11 stages. It should be noted that starting from the rightmost digit can be used to determine whether a number can be divided by 11. It is not required to start from the leftmost digit.

Steps in Rule of Divisibility of 11

Step 1: Begin from the leftmost or rightmost digit.
Step 2: Find the aggregate of all the digits in the odd locations.
Step 3: Find the aggregate of all the digits in the even locations
Step 4: Calculate the difference between the totals you got in steps 2 and 3.
Step 5: The difference must be zero or a value that 11 can divide exactly without producing a remainder, and then the number is completely divisible by 11.
The candidate can further go through the GMAT test-taking tips in order to solve the divisibility problems in a quicker and easier way.
A number is totally divisible by 11 if the difference between the sum of its alternative digits is divisible by 11. The process to determine whether a number, such as 2143, is divisible by 11 is listed below.

• Alternate digits, i.e., those in odd places together and those in even places together should be grouped. These two groups here are 24 and 13.
• Take the total of each group's digits, i.e., 2+4=6 and 1+3=4.
• Find the difference between the amounts now, 6-4=2.
• If the difference can be divided by 11, the original number can be as well. Here, the difference is 2, which cannot be divided by 11.
• 2143 is therefore not divisible by 11.

There are a couple more requirements to determine whether a number can be divided by 11. Here, they are described using the following examples:
If a number has an even number of digits, add the first digit and deduct the final digit from the remainder.

For Instance: 3784
There are four digits.
Now, 78 + 3 – 4 = 77 = 7 × 11
Thus, 3784 can be divided by 11.

There are several good GMAT books that would help the candidates to practice more topics regarding the divisibility rule. The candidates can follow the “GMAT Prep Plus 2022-2023” book to get familiar with similar types of topics to score better in the GMAT exam.

Rules:

1) The rule for the number 11 is quite straightforward: add the sum of the odd place digit to the total of the even place digit. When a number is split by 11, the original number is also divided by 11.
Now, the candidate may have encountered a number when practising this rule. It is initially believed by analysing the number that it will not divisible by 11. However, later by following the divisibility rule it is found that the number is divisible by 11.
Let’s analyse this question:
If 54328978014 is divided by 11?
Sum of the numbers in odd places= (5+3+8+7+0+4)= 27
Sum of the numbers in even places= (4+2+9+8+1)= 24
Now, 27-24 equals 3, which cannot be divided by 11.

Let’s have a look at other approaches to understand the rule of divisibility of 11.

Approach Solution 1:
Subtract the first and last digits from the rest of the integer if the number of digits is odd.

For instance, 82907
The number consists of 5 digits.
Now, 290 – 8 – 7 = 275 × 11
So, 82907 can be divided by 11.

Approach Solution 2:
From the number's right end digit to its left end digit, create groups of two digits, and then calculate the sum of the resultant groups. The number is divisible by 11 if the sum is a multiple of 11.

For instance: 3774 is not divisible by 11 since 37 + 74 = 111 and 1 + 11 = 12.
253 is divisible by 11 since 2 + 53 = 55 = 5 x 11.

Approach Solution 3:
Deduct the last digit of the number from the remaining number. The actual number will be divisible by 11 if the resultant number is a multiple of 11.

For instance: 9647
9647 = 964 – 7 = 957
957 = 95 – 7 = 88 = 8 × 11
Therefore, 9647 can be divided by 11.

Other Examples:

(i) 154

Sum of the numbers in an even position = 5.
Sum of the numbers in the odd position = 1 + 5 = 6
Difference between the two sums = 5 - 6 = -1
-1 can be divided by 11.
Therefore, 154 can be divided by 11.

(ii) 814

Sum of the numbers in an even position = 1
Sum of the numbers in the odd position = 8 + 4 = 12
Difference between the two sums = 1 - 12 = - 11.
-11 can be divided by 11.
Therefore, 814 can be divided by 11.

(iii) 957

Sum of the numbers in an even position = 5.
Sum of the numbers in the odd position = 9 + 7 = 16.
Difference between the two sums= 5 – 16 = –11.
-11 can be divided by 11.
Therefore, 957 can be divided by 11.

(iv) 1023

Sum of the numbers in an even position = 0 + 3 = 3.
Sum of the numbers in the odd position = 1 + 2 = 3.
The difference between the two sums is equal to zero (3-3)
11 can be divided by zero.
Therefore, 1023 can be divided by 11.

(v) 1122

Sum of the numbers in an even position = 1 + 2 = 3.
Sum of the numbers in the odd position = 1 + 2 = 3.
The difference between the two sums is equal to zero (3-3)
11 can be divided by zero.
Therefore, 1122 can be divided by 11.

(vi) 1749

Sum of the numbers in an even position = 7 + 9 = 16.
Sum of the numbers in the odd position = 1 + 4 = 5.
16 - 5 = 11 is the difference between the two sums.
11 can be divided by 11.
Therefore, 1749 can be divided by 11.

(vii) 53856

Sum of the numbers in an even position = 3 + 5, which is 8.
Sum of the numbers in the odd position = 5 + 8 + 6, which is 19.
8 - 19 = -11 is the difference between the two sums.
-11 can be divided by 11.
Therefore, 53856 can be divided by 11.

(viii) 592845

Sum of the numbers in an even position = 9 + 8 + 5 = 22.
Sum of the numbers in the odd position = 5 + 2 + 4 = 11
22 - 11 = 11 is the difference between the two sums.
11 can be divided by 11.
Therefore, 592845 can be divided by 11.

(ix) 5048593

Sum of the numbers in an even position = 0 + 8 + 9 = 17.
Sum of the numbers in the odd position = 5 + 4 + 5 + 3 = 17.
17 - 17 = 0 is the difference between the two sums.
11 can be divided by zero.
Therefore, 5048593 can be divided by 11.

(x) 98521258

Sum of the numbers in an even position = 8 + 2 + 2 + 8 = 20.
Sum of the numbers in the odd position = 9 + 5 + 1 + 5 and equal 20.
20 - 20 = 0 is the difference between the two sums.
11 can be divided by zero.
98521258 can be divided by 11 thus.

The candidates can follow GMAT Quant practice papers to get problems regarding such divisibility rules. The GMAT Quant section enables the candidates to improve their mathematical learning and enhance their calculative knowledge.

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