6. There is a triple $k, m, n$ of positive integers without common factors such that $k \log _{400} 5+m \log _{400} 2=n$. Find the sum $k+m+n$.
$\begin{array}{l}\text { (ans. }(k, m, n)=(2,4,1), k+m+n=7 . \\ \log _{400} 5^{k} 2^{m}=\log _{400} 400^{n} \Rightarrow 5^{k} 2^{m}=400^{n}=\left(2^{4} 5^{2}\right)^{n} \Rightarrow m=4 n, k= \\ 2 n \Rightarrow(k, m, n)=(2,4,1) .)\end{array}$