Find the value of A and B if `{:(" 41 A"),(+"B 4"),(ulbar(" 5 1 2")):}`

Answer

To solve the problem of finding the values of A and B in the equation \(41A + B4 = 512\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Equation**: We need to rewrite the numbers in a more manageable form. The number \(41A\) can be expressed as \(410 + A\) and \(B4\) can be expressed as \(10B + 4\). Thus, our equation becomes:
\[ (410 + A) + (10B + 4) = 512 \]

2. **Combine Like Terms**: Simplifying the equation gives:
\[ 410 + A + 10B + 4 = 512 \]
This simplifies to:
\[ A + 10B + 414 = 512 \]

3. **Isolate A and B**: Now, subtract 414 from both sides:
\[ A + 10B = 512 - 414 \]
This simplifies to:
\[ A + 10B = 98 \]

4. **Analyze the Units Digit**: Since \(A\) and \(B\) are single digits (0-9), we can analyze the equation \(A + 4\) (from \(B4\)) to find the units digit of the sum. The last digit of \(512\) is \(2\). Thus, we have:
\[ A + 4 \equiv 2 \mod 10 \]
This means:
\[ A + 4 = 2 \quad \text{or} \quad A + 4 = 12 \]
From \(A + 4 = 2\), we get \(A = -2\) (not valid). From \(A + 4 = 12\), we get:
\[ A = 8 \]

5. **Substitute A Back**: Now substitute \(A = 8\) back into the equation \(A + 10B = 98\):
\[ 8 + 10B = 98 \]
Subtract \(8\) from both sides:
\[ 10B = 90 \]
Divide by \(10\):
\[ B = 9 \]

6. **Final Values**: Therefore, the values of \(A\) and \(B\) are:
\[ A = 8, \quad B = 9 \]

### Summary:
The values are:
- \(A = 8\)
- \(B = 9\)