To determine the remainder when $293158020640006078477$ is divided by 11, we use the divisibility rule for 11.

Divisibility Rule for 11:

1. Starting from the rightmost digit, alternate between adding and subtracting the digits.
2. If the result is divisible by 11 (including 0), the original number is divisible by 11. Otherwise, the remainder when dividing the result by 11 gives the remainder of the original number.

Let’s apply this step-by-step.

The number is $293158020640006078477$.

We alternate adding and subtracting digits: $7 - 7 + 4 - 7 + 8 - 0 + 6 - 0 + 0 - 0 + 4 - 6 + 0 - 2 + 0 - 8 + 5 - 1 + 9 - 2 + 3 = -7$

Now, divide the result by 11: $-7 \div 11 = -1 \text{ remainder } 4$

Since the result is negative, add 11 to the remainder to get a positive remainder: $4 + 11 = 15 \text{ remainder } 4$

Therefore, the remainder when $293158020640006078477$ is divided by 11 is 4.

Would you like more details or have any questions? Here are some follow-up questions:

1. How does the divisibility rule for 7 differ from that for 11?
2. Can you show an example of a number divisible by 11?
3. What is the general formula for divisibility by a prime number?
4. How would you compute the remainder of a large number divided by 7?
5. How does the remainder change if the last digit of the number is altered?

Tip: For very large numbers, divisibility rules provide an efficient way to check without fully performing the division.