To find the number of three digits that leaves the same remainder when divided by 11284 and 7655, we need to find the greatest common divisor (GCD) of these two numbers.

1. Finding the GCD:
We can use the Euclidean algorithm to find the GCD of 11284 and 7655. The algorithm states that the GCD of two numbers is equal to the GCD of the smaller number and the difference between the larger and smaller numbers.

Using this algorithm:
GCD(11284, 7655) = GCD(7655, 11284 - 7655)
= GCD(7655, 3629)
= GCD(3629, 7655 - 2*3629)
= GCD(3629, 3943)
= GCD(3943, 3629 - 3943)
= GCD(3943, -314)

Since the second number is negative, we can ignore the sign and continue the algorithm:
GCD(3943, 314) = GCD(314, 3943 - 12*314)
= GCD(314, 5)
= GCD(5, 314 - 62*5)
= GCD(5, 4)
= GCD(4, 5 - 4*1)
= GCD(4, 1)
= GCD(1, 4 - 4*1)
= GCD(1, 0)

The GCD(1, 0) is 1, so the GCD of 11284 and 7655 is 1.

2. Finding the number of three digits:
Since the GCD of 11284 and 7655 is 1, any three-digit number greater than 1 can be a possible answer. However, we need to choose the smallest three-digit number that leaves the same remainder when divided by both 11284 and 7655.

The smallest three-digit number is 100. We can check if it leaves the same remainder when divided by 11284 and 7655:
100 % 11284 = 100
100 % 7655 = 100

Therefore, the number of three digits is 100.

3. Finding their sum:
To find the sum of the three digits, we add them together:
1 + 0 + 0 = 1

Therefore, the sum of the three digits is 1.

Conclusion:
The number of three digits that leaves the same remainder when divided by 11284 and 7655 is 100, and their sum is 1. Hence, the correct option is A) 191 11.