If A, B, and C stand for different digits, and BA+AC=BCA, what digits do A, B, and C represent?

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To find the values of A, B, and C that satisfy the equation BA + AC = BCA, we need to approach the problem systematically.

1. Start by breaking down the numbers in the equation into their corresponding place values:
- BA = 10B + A
- AC = 10A + C
- BCA = 100B + 10C + A

2. Substitute the expressions into the equation:
10B + A + 10A + C = 100B + 10C + A

3. Simplify the equation by combining like terms:
11A + C = 99B + 9C

4. Rearrange the equation to isolate B:
99B = 11A + C - 9C - 11A
99B = -8C

5. Since B represents a digit, it must be a single digit. Therefore, the only possible value for B is 0.

6. Substitute B = 0 back into the equation:
0 = -8C

7. Solve for C:
-8C = 0
C = 0

8. Finally, substitute B = 0 and C = 0 back into the equation to find A:
11A + 0 = 11A = 0
A = 0

Therefore, A = 0, B = 0, and C = 0.