Question

There exist positive integers $A,B$ and $C$ with no common factors greater than 1 , such that A ${\log }_{200}5+B{\log }_{200}2=$ C. The sum $(A+B+C)$ equals

• A. 5
• B. 6
• C. 7
• D. 8

Answer 1

Answer: (B)

$AC{\log }_{200}5+B{\log }_{200}2=C$
$\frac{A\log 5}{\log 200}+\frac{B\log 2}{\log 200}=C$
$A\log 5+B\log 2=C\log 200=C\log \left(5^2{2}^3\right)=2C\log 5+3C\log 2$
$\text{ hence, }A=2C$
$B=3C$
for no common factor greater than $1,C=1$
$\therefore A=2;B=3\Rightarrow A+B+C=6$