To find the sum of the digits of the resulting number when 111111111 is multiplied by itself, we first calculate the product:

•   11 × 11 = 121

•   111 × 111 = 12321

•   1111 × 1111 = 1234321

•   Following the same pattern, we get: (1111 …. 9 times) × (1111 …. 9 times) = 12345678987654321

•   Sum of the digits of the resulting number = 2 × (1 + 2 + 3 + …. + 8) + 9 = 72 + 9 = 81

•   Hence, option (c) is correct.

Using Formula:

•   We can use this formula also: Sum of first n natural numbers = n (n + 1) / 2

•   Sum of the digits of the resulting number = 2 × (8 × 9/2) + 9 = 72 + 9 = 81

Approach 2:

•   Each digit of the 9-digit number is 1, so the sum of its digits is 9. (1+1+1+1+1+1+1+1+1=9).

•   This means the number is divisible by 9.

•   Any multiple of this number will also be divisible by 9.

•   Therefore, the sum of the digits of the resulting number from the multiplication of 111111111 by 111111111 must also be divisible by 9.

•   The correct answer will be the option that is a multiple of 9, which is 81.

•   Therefore, option (c) is correct.