How many three-digit even numbers which are divisible by 11 also have the sum of their digits equal to 10?

To find the three-digit even numbers that are divisible by 11 and have the sum of their digits equal to 10, let's analyze the problem step-by-step.

Let:
- The three-digit number be represented as \( \overline{abc} = 100a + 10b + c \), where \( a \), \( b \), and \( c \) are the hundreds, tens, and units digits respectively.
- The conditions are:
  1. \( \overline{abc} \) is even, so \( c \) must be even.
  2. \( \overline{abc} \) is divisible by 11.
  3. The sum of the digits \( a + b + c = 10 \).

### Step 1: Condition for Divisibility by 11
For a number to be divisible by 11, the alternating sum of its digits must be divisible by 11. This means:
\[
a - b + c = 0 \quad \text{or} \quad a - b + c = 11
\]

### Step 2: Finding Possible Values of \( a \), \( b \), and \( c \)
Since \( \overline{abc} \) is a three-digit number, \( a \) must be between 1 and 9. Additionally, since \( \overline{abc} \) is even, \( c \) must be an even digit (0, 2, 4, 6, or 8).

We will now use both conditions to identify valid values for \( a \), \( b \), and \( c \) such that:
1. \( a + b + c = 10 \)
2. \( a - b + c = 0 \) (since \( a - b + c = 11 \) would exceed the digit constraint)

### Step 3: Solving the Equations
Using the system:
\[
a + b + c = 10
\]
\[
a - b + c = 0
\]

Add the two equations:
\[
2a + 2c = 10 \implies a + c = 5
\]

Substitute \( a + c = 5 \) back into the first equation:
\[
a + b + c = 10 \implies 5 + b = 10 \implies b = 5
\]

Thus, we have:
1. \( b = 5 \)
2. \( a + c = 5 \)

Now, we examine possible pairs \((a, c)\) where \( a + c = 5 \) and \( c \) is even:
- \( a = 1 \), \( c = 4 \)
- \( a = 3 \), \( c = 2 \)
- \( a = 5 \), \( c = 0 \)

### Step 4: Constructing the Numbers
The possible three-digit numbers \(\overline{abc}\) that meet all conditions are:
- \(154\) (with \( a = 1 \), \( b = 5 \), \( c = 4 \))
- \(352\) (with \( a = 3 \), \( b = 5 \), \( c = 2 \))
- \(550\) (with \( a = 5 \), \( b = 5 \), \( c = 0 \))

### Step 5: Verification
Each of these numbers:
1. Is even.
2. Has digits summing to 10.
3. Is divisible by 11.

Therefore, there are **3 such numbers: 154, 352, and 550**.

### Final Answer
There are **3 three-digit even numbers** that are divisible by 11 and have the sum of their digits equal to 10.

Would you like a more detailed breakdown or have any questions about the steps?

---

1. How would the solution change if the sum of the digits were 12 instead of 10?
2. Can this method be applied to four-digit numbers, and what modifications would be needed?
3. What would the solution be if the numbers had to be odd instead of even?
4. How would the solution differ if we considered divisibility by another number, like 7?
5. How can we generalize this method to find numbers with different divisibility and sum conditions?

**Tip:** Always check conditions one by one to simplify complex constraints.

Explore how to find three-digit even numbers divisible by 11 with a digit sum of 10. This problem involves number theory concepts, divisibility rules, and step-by-step calculations, making it suitable for Grades 9-10 students.

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