6. Among the first 999 positive integers, the numbers that are neither divisible by 5 nor by 7 are
$\qquad$ in number.
6. 686 .

Let $I=\{1,2,3, \cdots, 999\}, A=\{x \mid x \in I$, and $5 \mid x\}, B=\{x \mid x \in I$, and $7 \mid x\}$, then $|I|=999,|A|=\left[\frac{999}{5}\right]=199|B|=\left[\frac{999}{7}\right]=142,|A \cap B|=\left[\frac{999}{5 \times 7}\right]=28$.
Therefore, $|\overline{A \cup B}|=|I|-(|A|+|B|)+|A \cap B|=686$.