In the equation AB + BA = AAC, what is the primary requirement for the digits A, B, and C?

In the equation AB + BA = AAC, where AB and BA represent two-digit numbers constructed from the digits A and B, the primary requirement is that A, B, and C must be different digits.

To clarify this, let's analyze the structure of the equation. The two-digit numbers AB and BA can be expressed as 10A + B and 10B + A, respectively. When we add these two expressions, we arrive at:

(10A + B) + (10B + A) = 11A + 11B = 11(A + B).

Since the result is expressed as AAC, the number AAC can be translated to 100A + 10A + C, simplifying to 110A + C.

Setting these equal gives us the equation:

11(A + B) = 110A + C.

For this equality to hold true, it requires distinct values for A, B, and C because each corresponds to separate digits in the two-digit and three-digit numbers presented. If any of the digits were the same, it would violate the principles of unique representation in the positional numeral system.

Therefore, the requirement that A, B, and C must be different digits is essential for the equation to function