Find the values of A,B,C when `({:(," "AB),(,""xxBA):})/(" "BCB)`

Answer

To solve the equation \((AB) \times (BA) = BCB\), we need to find the values of \(A\), \(B\), and \(C\). Let's break down the steps:

### Step 1: Understand the Equation
The equation states that when we multiply \(AB\) (a two-digit number where \(A\) is the tens digit and \(B\) is the units digit) by \(BA\) (where \(B\) is the tens digit and \(A\) is the units digit), we should get \(BCB\) (a three-digit number where \(B\) is the hundreds and units digit, and \(C\) is the tens digit).

### Step 2: Express the Numbers Mathematically
We can express the numbers as:
- \(AB = 10A + B\)
- \(BA = 10B + A\)
- \(BCB = 100B + 10C + B = 101B + 10C\)

### Step 3: Set Up the Equation
Now we can set up the equation based on the multiplication:
\[ (10A + B)(10B + A) = 101B + 10C \]

### Step 4: Expand the Left Side
Expanding the left side:
\[ (10A + B)(10B + A) = 100AB + 10A^2 + 10B^2 + AB = 100AB + 10A^2 + 10B^2 + AB = 101AB + 10A^2 + 10B^2 \]

### Step 5: Set the Equation Equal
Now we have:
\[ 101AB + 10A^2 + 10B^2 = 101B + 10C \]

### Step 6: Rearrange the Equation
Rearranging gives us:
\[ 101AB + 10A^2 + 10B^2 - 101B - 10C = 0 \]

### Step 7: Substitute Possible Values
Since \(A\), \(B\), and \(C\) are digits (0-9), we can substitute possible values for \(A\) and \(B\) to find \(C\).

### Step 8: Testing Values
Let's test \(A = 1\) and \(B = 1\):
\[ AB = 11, \quad BA = 11 \quad \Rightarrow \quad 11 \times 11 = 121 \]
This gives \(BCB = 121\), which implies \(C = 2\).

### Step 9: Conclusion
Thus, we find:
- \(A = 1\)
- \(B = 1\)
- \(C = 2\)

### Final Answer
The values are:
- \(A = 1\)
- \(B = 1\)
- \(C = 2\)

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