In each of the following replace * by a digit so that the number formed is divisible by 11:

(i) 8*9484

(ii) 9*53762

A number is divisible by 11 if the difference between the sum of digits at odd places (from ones) and the sum of digits at even places (from tens) is either 0 or divisible by 11.

(i) 8*9484

Digits from right: 4, 8, 4, 9, *, 8.

Sum of digits at odd places (from right) = 4 + 4 + * = 8 + *.

Sum of digits at even places (from right) = 8 + 9 + 8 = 25.

Difference = 25 − (8 + *) = 17 − *.

For divisibility by 11, 17 − * = 0 (impossible) or 17 − * = 11 ⇒ * = 6.

Hence, * should be replaced by 6. The number formed is 869484.

(ii) 9*53762

Digits from right: 2, 6, 7, 3, 5, *, 9.

Sum of digits at odd places (from right) = 2 + 7 + 5 + 9 = 23.

Sum of digits at even places (from right) = 6 + 3 + * = 9 + *.

Difference = 23 − (9 + *) = 14 − *.

For divisibility by 11, 14 − * = 0 (impossible) or 14 − * = 11 ⇒ * = 3.

Hence, * should be replaced by 3. The number formed is 9353762.