Question

In each of the following numbers. replace * by the smallest number to make it divisible by 11:

(i) 86 * 72
(ii) 467 * 91
(iii) 9 * 8071

Answer 1

Rule: A number is divisible by 11 if the difference of the sums of the alternate digits is either 0 or a multiple of 11.

(i) 86 $\times$ 72

Sum of the digits at the odd places = 8 + missing number + 2 = missing number + 10
Sum of the digits at the even places = 6 + 7 = 13
Difference = [missing number + 10 ] − 13 = Missing number − 3
According to the rule, missing number − 3 = 0 [∵ the missing number is a single digit]
Thus, missing number = 3
Hence, the smallest required number is 3.

(ii) 467 $\times$ 91

Sum of the digits at the odd places = 4 + 7 + 9 = 20
Sum of the digits at the even places = 6 + missing number + 1 = missing number + 7
Difference = 20 − [missing number + 7] = 13 − missing number
According to rule, 13 − missing number = 11 [∵ the missing number is a single digit]
Thus, missing number = 2
Hence, the smallest required number is 2.

(iii) 9 $\times$ 8071

Sum of the digits at the odd places = 9 + 8 + 7 = 24
Sum of the digits at the even places = missing number + 0 + 1 = missing number + 1
Difference = 24 − [missing number + 1] = 23 − missing number
According to rule, 23 − missing number = 22 [∵ 22 is a multiple of 11 and the missing number is a single digit]
Thus, missing number = 1
Hence, the smallest required number is 1.