To solve this problem, we need to find the remainder when the number formed by repeating the digits "147" a total of 841 times is divided by 11. This number can be expressed as:

$N = 147147147\ldots147 \quad (\text{repeated 841 times}).$

Step 1: Understanding the divisibility rule for 11

The divisibility rule for 11 states that a number is divisible by 11 if the alternating sum of its digits (starting from the left) is divisible by 11. This alternating sum is computed by subtracting and adding successive digits. So, for a number like $147$, we compute the alternating sum as:

$S(147) = 1 - 4 + 7 = 4.$

Step 2: Computing the alternating sum for the repeated number

Since $N$ is formed by repeating the block "147" exactly 841 times, the alternating sum for the entire number will just be the alternating sum of "147" repeated 841 times. Therefore, the alternating sum of $N$ is:

$S(N) = 841 \times S(147) = 841 \times 4 = 3364.$

Step 3: Finding the remainder when divided by 11

Now, we need to find the remainder when 3364 is divided by 11. We can perform this division:

$3364 \div 11 = 305 \text{ remainder } 9.$

Thus, the remainder when the number formed by repeating "147" 841 times is divided by 11 is 9.

Conclusion:

The remainder when the number is divided by 11 is $\boxed{9}$.

Would you like more details or have any questions on this? Here are some related questions:

1. What is the general rule for divisibility by 11 for large numbers?
2. How can you compute remainders for numbers formed by repeating a digit pattern?
3. What happens when a number like "147" is repeated an even number of times for divisibility by 11?
4. Can this method be extended to find remainders with other divisors like 7 or 13?
5. How does modular arithmetic simplify large number division problems?

Tip: When working with large numbers, always check if patterns like alternating sums or digit repetition can simplify the problem.