To solve this problem, we need to find three-digit even numbers that meet two criteria:

1. They are divisible by 11.
2. The sum of their digits equals 10.

Step 1: Understanding divisibility by 11

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 a multiple of 11 (including 0). For a three-digit number represented as $ABC$, where $A$, $B$, and $C$ are its hundreds, tens, and units digits respectively, the condition for divisibility by 11 is: $A - B + C \equiv 0 \pmod{11}.$

Step 2: Sum of digits condition

We need the sum of the digits to be: $A + B + C = 10.$

Step 3: Constraints for an even number

Since the number is even, the units digit $C$ must be an even number (0, 2, 4, 6, or 8).

Step 4: Solving the problem

Let’s explore possible values of $A$, $B$, and $C$ that satisfy all these conditions.

Case by case analysis for different even values of $C$:

1. If $C = 0$:

   • Then $A + B = 10$.
   • $A - B + 0 = 0 \rightarrow A - B = 0 \rightarrow A = B$.
   • The only solution is $A = B = 5$, giving the number 550.

2. If $C = 2$:

   • Then $A + B = 8$.
   • $A - B + 2 = 0 \rightarrow A - B = -2 \rightarrow A = B - 2$.
   • Possible values are $A = 3$, $B = 5$, giving the number 352.

3. If $C = 4$:

   • Then $A + B = 6$.
   • $A - B + 4 = 0 \rightarrow A - B = -4 \rightarrow A = B - 4$.
   • Possible values are $A = 1$, $B = 5$, giving the number 154.

4. If $C = 6$:

   • Then $A + B = 4$.
   • $A - B + 6 = 0 \rightarrow A - B = -6 \rightarrow A = B - 6$.
   • Possible values are $A = 1$, $B = 7$, giving the number 176.

5. If $C = 8$:

   • Then $A + B = 2$.
   • $A - B + 8 = 0 \rightarrow A - B = -8 \rightarrow A = B - 8$.
   • There are no valid solutions for $A$ and $B$ here.

Final Valid Numbers

The three-digit even numbers that satisfy all conditions are:

• 550
• 352
• 154
• 176

Thus, there are four such numbers.

Would you like more details or have any questions?

Related Questions:

1. How many three-digit numbers are divisible by 11?
2. How can we solve this problem using a different method, such as modular arithmetic?
3. What would change if the sum of digits condition was different, for example, equal to 12?
4. How can the divisibility rule for 11 be generalized to four-digit numbers?
5. What are the properties of numbers that are both even and divisible by another factor (e.g., 7)?

Tip:

When solving problems with multiple conditions, try to break down each condition separately and find the intersection of all possible solutions.