8.32 If the integers $A, B, C$ are relatively prime and satisfy $A \log _{200} 5+B \log _{200} 2=C$, then $A+B+C=$
(A) 6 .
(B) 7 .
(C) 8 .
(D) 9 .
(E) 10 .
(46th American High School Mathematics Examination, 1995)
[Solution] From the given, $\log _{200} 5^{A}+\log _{200} 2^{B}=C$, we know $5^{A} \cdot 2^{B}=200^{C}$, and $200^{C}=(8 \times 25)^{C}=2^{3 C} \cdot 5^{2 C}$. Thus, $A=2 C, B=3 C$, and since $(A, B, C)=1$, $\therefore C=1, A=2, B=3$. Therefore, $A+B+C=6$. Hence, the answer is $(A)$.