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The Divisibility Rule of 11 states that a number is divisible by 11 only if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either 0 or a multiple of 11.
For example, let’s consider the number 3258. The sum of its digits in odd positions (5 and 3) is 8, while the sum of its digits in even positions (8 and 2) is 10. The difference between these two sums is 2. Since 2 is not a multiple of 11, we can conclude that 5832 is not divisible by 11.
Overall, the divisibility rule of 11 can be useful for quickly determining whether a number is divisible by 11 without having to perform long division.
Divisibility Rules
Divisibility rules are a set of rules that help us determine whether a given number is divisible by another number without actually dividing it. These rules are helpful in simplifying calculations and reducing the time taken to solve problems.
One such rule is the divisibility rule of 11, which helps us determine whether a given number is divisible by 11. In this article, we will explore the divisibility rule of 11 in detail and provide solved examples to understand the application of the divisibility rule of 11.
Basis of the Divisibility Rule of 11
The divisibility rule of 11 is based on the fact that any number can be expressed as a sum of powers of 10. For example, the number 5832 can be expressed as
5 x 10^{3} + 8 x 10^{2} + 3 x 10^{1} + 2 x 10^{0}
We can observe that the powers of 10 alternate between odd and even positions. The position of the first digit is considered to be odd. In the case of 5832, the digit 5 is in the odd position, the digit 8 is in the even position, the digit 3 is in the odd position, and the digit 2 is in the even position.
Divisibility Rule of 11
The divisibility rule of 11 states that a number is divisible by 11 only if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either 0 or a multiple of 11.
Now, let’s check if 231 is divisible by 11. Precisely, the sum of the digits in odd positions (1 and 2) is 3, while the sum in even positions (3) is 3. The difference between these two sums is 0. Therefore, we can conclude that 231 is divisible by 11.
Example: Is 9768 divisible by 11?
Solution: The sum of the digits in odd positions (9 and 7) is 16, while the sum in even positions (8 and 6) is 14. The difference between these two sums is 2. Since 2 is not a multiple of 11, 9768 is not divisible by 11.
Advantages of the Divisibility Rule of 11
The divisibility rule of 11 has several advantages:
 Saves time
The rule helps us quickly determine whether a given number is divisible by 11 without performing long division.
 Easy to apply
The rule is easy to apply and requires only basic arithmetic operations.

Applicable to large numbers
The rule can be applied to numbers of any size, making it useful in a wide range of applications.
 Helps in problemsolving
The rule can be used to solve problems related to divisibility by 11, such as finding the smallest number that is divisible by both 11 and another number.
Divisibility Rule of 11: Conclusion
The divisibility rule of 11 is useful for quickly determining whether a given number is divisible by 11. The rule is based on the fact that any number can be expressed as a sum of powers of 10, and the difference between the sum of its digits in odd positions and the sum of its digits in even positions is either 0 or a multiple of 11.
The rule has several advantages, including saving time, being easy to apply, applying to large numbers, and helping in problemsolving.
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FAQs on Divisibility Rule of 11
How does the Divisibility Rule of 11 work?
The Divisibility Rule of 11 is based on the alternating pattern of digits in odd and even positions when a number is expressed as a sum of powers of 10. To determine if a number is divisible by 11, you calculate the difference between the sum of digits in odd positions and the sum of digits in even positions. If this difference is either 0 or a multiple of 11, then the number is divisible by 11.
Can the Divisibility Rule of 11 be applied to numbers with decimals?
No, the Divisibility Rule of 11 is specifically designed for integers. It doesn't apply to numbers with decimal fractions.
Does the rule work for negative numbers?
Yes, the Divisibility Rule of 11 works for negative numbers as well. When dealing with negative integers, treat the minus sign as a regular digit and apply the rule as usual.
Can the Divisibility Rule of 11 be used to find the smallest number divisible by both 11 and another?
Yes, the rule can be combined with other divisibility rules to find the smallest number divisible by both 11 and another given number. By applying the rules together, you can efficiently find such numbers.
Can I use the Divisibility Rule of 11 to check if a number is prime?
No, the Divisibility Rule of 11 is not applicable for determining prime numbers. It can only tell you whether a number is divisible by 11, not whether it is prime or has other factors. There are other methods and tests, such as the Sieve of Eratosthenes or the primality test, to check for prime numbers.