6. Let the function $f(x)=\sin ^{4} \frac{k x}{10}+\cos ^{4} \frac{k x}{10}$, where $k$ is a positive integer. If for any real number $a$, we have $\{f(x) \mid a<x<a+1\}=\{f(x) \mid x \in \mathbf{R}\}$, then the minimum value of $k$ is $\qquad$.
$$
\begin{array}{l}
f(x)=\sin ^{4} \frac{k x}{10}+\cos ^{4} \frac{k x}{10}=1-2 \sin ^{2} \frac{k x}{10} \cos ^{2} \frac{k x}{10}=1-\frac{1}{2} \sin ^{2} \frac{k x}{5} \\
=1-\frac{1}{4}\left(1-\cos \frac{2 k x}{5}\right)=\frac{3}{4}+\frac{1}{4} \cos \frac{2 k x}{5} .
\end{array}
$$

According to the problem, the smallest positive period of $f(x)$, $T=\frac{5 \pi}{k}5 \pi \Rightarrow k \geqslant 16$. Therefore, the minimum value of $k$ is 16.