The question asks how many numbers from the given set: $2037, 13754, 109911, 5863, 85657, 567952$ are divisible by 11.

To solve this, we will apply the divisibility rule for 11:

• The rule states that a number is divisible by 11 if the difference between the sum of its digits in odd positions and the sum of its digits in even positions is divisible by 11 (or zero).

Let's check each number:

1. 2037:

   • Odd positions: $2 + 3 = 5$
   • Even positions: $0 + 7 = 7$
   • Difference: $5 - 7 = -2$, not divisible by 11.

2. 13754:

   • Odd positions: $1 + 7 + 4 = 12$
   • Even positions: $3 + 5 = 8$
   • Difference: $12 - 8 = 4$, not divisible by 11.

3. 109911:

   • Odd positions: $1 + 9 + 1 = 11$
   • Even positions: $0 + 9 + 1 = 10$
   • Difference: $11 - 10 = 1$, not divisible by 11.

4. 5863:

   • Odd positions: $5 + 6 = 11$
   • Even positions: $8 + 3 = 11$
   • Difference: $11 - 11 = 0$, divisible by 11.

5. 85657:

   • Odd positions: $8 + 6 + 7 = 21$
   • Even positions: $5 + 5 = 10$
   • Difference: $21 - 10 = 11$, divisible by 11.

6. 567952:

   • Odd positions: $5 + 7 + 5 = 17$
   • Even positions: $6 + 9 + 2 = 17$
   • Difference: $17 - 17 = 0$, divisible by 11.

So, the numbers divisible by 11 are 5863, 85657, 567952. Thus, the correct answer is 3 numbers.

Answer: B. 3

Would you like further details or explanations on any part of this process?

Here are 5 related questions for you:

1. How can we generalize the rule for divisibility by other numbers, such as 7 or 13?
2. What other numbers can you check for divisibility rules using similar methods?
3. Can you create a program to automate checking divisibility for large lists of numbers?
4. What are the properties of numbers that are divisible by 11 in terms of digit patterns?
5. How is divisibility by 11 related to modular arithmetic?

Tip: For large numbers, it can be helpful to use a calculator or a script to check divisibility rules quickly.