G4.3 Given that $A, B, C$ are positive integers with their greatest common divisor equal to 1 . If $A, B, C$ satisfy $A \log _{500} 5+B \log _{500} 2=C$, find the value of $A+B+C$.
$$
\log _{500} 5^{A}+\log _{500} 2^{B}=\log _{500} 500^{C} \Rightarrow \log _{500} 5^{A} \cdot 2^{B}=\log _{500} 500^{C} \Rightarrow 5^{A} \cdot 2^{B}=5^{3 C} \cdot 2^{2 C}
$$
$A=3 C, B=2 C$ (unique factorization theorem)
$\because A, B, C$ are relatively prime.
$$
\begin{array}{l}
\therefore C=1, A=3, B=2 \\
A+B+C=6
\end{array}
$$