A set of specific rules used to check whether a given number is divisible by a divisor without performing the entire division procedure is called the divisibility rule. The divisibility by 11 rule states that if the difference between the sum of the alternative digits (sum of all odd places digits and the sum of all even places digits) is divisible by 11 or 0, then the provided number is also divisible by 11. Solution: We know that a number is divisible by 11 if the difference of the sum of alternate digits is 0 or a multiple of 11. (i)86 * 72 Sum of digits at odd places = 8 + missing number + 2 = 10 + missing number Sum of digits at even places = 6 + 7 = 13 Difference between them = 10 + missing number – 13 So the difference = missing number – 3 We know that missing number – 3 = 0 as it is a single digit Missing number = 3 Therefore, the smallest required number to make it divisible by 11 is 3. (ii)467 * 91 Sum of digits at odd places = 4 + 7 + 9 = 20 Sum of digits at even places = 6 + missing number + 1 = 7 + missing number Difference between them = 20 – (7 + missing number) So the difference = 13 – missing number We know that 13 - missing number = 11 Missing number = 2 Therefore, the smallest required number to make it divisible by 11 is 2. (iii)9 * 8071 Sum of digits at odd places = 9 + 8 + 7 = 24 Sum of digits at even places = missing number + 0 + 1 = 1 + missing number Difference between them = 24 – (1 + missing number) So the difference = 23 – missing number We know that 23 – missing number = 22 Missing number = 1 Therefore, the smallest required number to make it divisible by 11 is 1.
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