Divisibility by 11

Divisibility Test

Divisibility tests, often known as division rules in Math, are used to see if a number is divisible by another integer without having to utilize the division method. If a number is completely divisible by another integer, the quotient will be a whole number and the remainder will be zero. A nonzero integer m divides an integer n if there is an integer q such that n=mq. To state that m is a divisor of n and that m is a factor of n, we use the notation m|n.

General Properties of Divisibility

Property 1 If a number is divisible by another number, it must also be divisible by all of its factors.

Example We know that 36 can be divided by 12. 1, 2, 3, 4, 6, 12 are all factors of 12. A number must be divisible by the product of two co-prime integers if it is divisible by each of them.

Property 2 If a number is divisible by each of two co-prime integers, it must also be divisible by the product of the two numbers.

Example We know that each of the numbers 5 and 8 divides 4320. Also 5 and 8 co primes. The number 4320 must be divisible by 40. We discover that it is correct by long division. 

Property 3 If a number is a factor of both of the given numbers, it must also be a factor of their total. 

Example We know that each of the numbers 49 and 63 has a factor of 7. As a result, 7 must be a factor of (49+63) = 112 to be correct. Clearly, 7 divides 112. 

Property 4 If a number is a factor of both the given numbers, it must also be a factor of the difference between them. 

Example We know that each of the numbers 65 and 117 has a factor of 13. As a result, 13 has to be a factor of (117-65) = 52. Clearly, 13 divides 52 exactly.

Test of Divisibility by 11

To find if a given number is divisible by 11 we take the first digit from the left to the right of a number and add an additional symbol to its left. Then subtract the result by the next digit, add the result by the third digit, subtract the result again by the fourth digit, and so on. The original number is divisible by 11 if the response is divisible by 11. From left to right, alternately add and subtract the digits of an integer. The original number is divisible by 11 if the response is divisible by 11. Make the alternating sum of a number’s digits. The number is divisible by 11 if the result is a multiple of 11. For Example, we check if 764852 is divisible by 11. So we take the sum of digits of odd places from the left = 7 + 4 + 5 = 16. And we take the sum of digits at even places from the left = 6 + 8 +2 = 16. The difference between the sum of digits at odd and even place = 16 – 16,  which is 0. Hence, 764852 is divisible by 11.

Conclusion

The divisibility rule of 11 helps us determine if any given number is divisible by 11 which means that 11 completely divides the given number giving the remainder zero. We determine this by applying the rule in which we check that the difference between the sum of digits at the odd number of places and the sum of digits at the even number of places is divisible by 11.