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

Options

A. `4, 8, 3`
B. `1, 2, 5`
C. `2, 7, 9`
D. `3, 8, 7`

Answer

To solve the problem of finding the values of A, B, and C in the equation \((AB) \times (BA) = BCB\), we can follow these steps:

### Step 1: Understand the multiplication
The expression can be interpreted as:
\[ (AB) \times (BA) = BCB \]
Where \(AB\) represents the two-digit number formed by A and B, and \(BA\) represents the two-digit number formed by B and A.

### Step 2: Rewrite the numbers
We can express \(AB\) and \(BA\) in terms of A and B:
\[ AB = 10A + B \quad \text{and} \quad BA = 10B + A \]
Thus, the multiplication becomes:
\[ (10A + B) \times (10B + A) = BCB \]

### Step 3: Expand the multiplication
Now, we expand the left side:
\[ (10A + B)(10B + A) = 100AB + 10A^2 + 10B^2 + AB = 100AB + 10A^2 + 10B^2 \]
This simplifies to:
\[ AB(100 + 1) + 10(A^2 + B^2) \]
So, we have:
\[ 101AB + 10(A^2 + B^2) \]

### Step 4: Express BCB
Now, let's express \(BCB\):
\[ BCB = 100B + 10C + B = 101B + 10C \]

### Step 5: Set the equations equal
Now we can set the two expressions equal to each other:
\[ 101AB + 10(A^2 + B^2) = 101B + 10C \]

### Step 6: Solve for A, B, and C
From the equation, we can see that we need to find values for A, B, and C that satisfy this equation.

1. Since \(B\) cannot be equal to \(A\), we can start by assuming values for \(B\) and finding corresponding values for \(A\) and \(C\).
2. We can also note that \(A\) must be a single-digit number (0-9), and \(B\) must also be a single-digit number (0-9).

### Step 7: Testing values
Let's test some values:
- If \(B = 2\):
- Then \(A\) can be \(1\) (since \(A\) cannot be equal to \(B\)).
- Now we calculate \(C\):
\[ C = 1 + B^2 = 1 + 2^2 = 1 + 4 = 5 \]
- Therefore, \(A = 1\), \(B = 2\), and \(C = 5\).

### Conclusion
The values are:
- \(A = 1\)
- \(B = 2\)
- \(C = 5\)