+39 If A, B, and C represent three distinct digits from 1 to 9 and they satisfy the following equations, what is the value of the sum A + B + C? (In the equation below, AA represents a two-digit number both of whose digits are A)
\(A+B=C\)
\(AA-B=2 \cdot C\)
\(C\cdot B = AA+A\)
+893 Let's analyze the equations one by one:
A + B = C: This tells us that C must be greater than both A and B.
AA - B = 2C: This equation can be rewritten as 10A + A - B = 2C, or 11A - B = 2C. Since C is greater than A and B, 2C must be greater than 11A.
This is only possible if A is a very small number.
C * B = AA + A: This equation can be rewritten as CB = 11A. Since 11A is a multiple of 11, CB must also be a multiple of 11.
Considering these three equations, we can deduce the following:
A must be 1: This is the only way to satisfy the second equation, 11A - B = 2C, with C being greater than A and B.
C must be 3: Since A is 1, the first equation becomes 1 + B = C. The only way to satisfy this with C being greater than B is if C is 3.
B must be 2: From the first equation, if A is 1 and C is 3, then B must be 2.
Therefore, A + B + C = 1 + 2 + 3 = 6.
