Answer:
There are 26 such numbers.
Step-by-step explanation:
To find three-digit balanced numbers divisible by 18, we need to consider the possible values of the middle digit and ensure that the sum of the three digits is divisible by 9 (since 18 is divisible by 9).
Let the three-digit number be \(abc\), where \(a\), \(b\), and \(c\) are the hundreds, tens, and units digits, respectively.
Given that \(b = \frac{a + c}{2}\), we can express \(b\) in terms of \(a\) and \(c\).
Now, we need to find the values of \(a\), \(b\), and \(c\) such that \(a + b + c\) is divisible by 9.
Let's go through the possible values:
1. If \(b = 0\), then \(a = c\), and \(a + b + c = 2a\).
- For \(a = 1\), \(c = 1\), \(abc = 101\).
- For \(a = 2\), \(c = 2\), \(abc = 202\).
- For \(a = 3\), \(c = 3\), \(abc = 303\).
- For \(a = 4\), \(c = 4\), \(abc = 404\).
- For \(a = 5\), \(c = 5\), \(abc = 505\).
- For \(a = 6\), \(c = 6\), \(abc = 606\).
- For \(a = 7\), \(c = 7\), \(abc = 707\).
- For \(a = 8\), \(c = 8\), \(abc = 808\).
- For \(a = 9\), \(c = 9\), \(abc = 909\).
2. If \(b = 1\), then \(a + c = 2\).
- Possible pairs are \(a = 1\), \(c = 1\), and \(a = 2\), \(c = 0\).
- This gives us \(110\) and \(201\).
3. If \(b = 2\), then \(a + c = 4\).
- Possible pairs are \(a = 1\), \(c = 3\), and \(a = 3\), \(c = 1\).
- This gives us \(213\) and \(312\).
4. If \(b = 3\), then \(a + c = 6\).
- Possible pairs are \(a = 1\), \(c = 5\), and \(a = 5\), \(c = 1\).
- This gives us \(315\) and \(513\).
5. If \(b = 4\), then \(a + c = 8\).
- Possible pairs are \(a = 1\), \(c = 7\), and \(a = 7\), \(c = 1\).
- This gives us \(417\) and \(714\).
6. If \(b = 5\), then \(a + c = 10\).
- Possible pairs are \(a = 1\), \(c = 9\), and \(a = 9\), \(c = 1\).
- This gives us \(519\) and \(915\).
7. If \(b = 6\), then \(a + c = 12\).
- Possible pairs are \(a = 3\), \(c = 9\), and \(a = 9\), \(c = 3\).
- This gives us \(639\) and \(936\).
8. If \(b = 7\), then \(a + c = 14\).
- Possible pairs are \(a = 5\), \(c = 9\), and \(a = 9\), \(c = 5\).
- This gives us \(759\) and \(957\).
9. If \(b = 8\), then \(a + c = 16\).
- Possible pairs are \(a = 7\), \(c = 9\), and \(a = 9\), \(c = 7\).
- This gives us \(879\) and \(978\).
So, the balanced three-digit numbers divisible by 18 are:
\[101, 202, 303, 404, 505, 606, 707, 808, 909, 110, 201, 213, 312, 315, 513, 417, 714, 519, 915, 639, 936, 759, 957, 879, 978\]
There are 26 such numbers.