1. Tardigrade
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  3. Mathematics
  4. 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

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Q. 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

Continuity and Differentiability

Solution:

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