8. (GBR 4) Let $S$ be the set of all the odd positive integers that are not multiples of 5 and that are less than $30 \mathrm{~m}, \mathrm{~m}$ being an arbitrary positive integer. What is the smallest integer $k$ such that in any subset of $k$ integers from $S$ there must be two different integers, one of which divides the other?
8. Take the subset $\left\{a_{i}\right\}=\{1,7,11,13,17,19,23,29, \ldots, 30 m-1\}$ of $S$ containing all the elements of $S$ that are not multiples of 3. There are 8 m such elements. Every element in $S$ can be uniquely expressed as $3^{t} a_{i}$ for some $i$ and $t \geq 0$. In a subset of $S$ with $8 m+1$ elements, two of them will have the same $a_{i}$, hence one will divide the other. On the other hand, for each $i=1,2, \ldots, 8 m$ choose $t \geq 0$ such that $10 \mathrm{~m}<$ $b_{i}=3^{t} a_{i}<30 \mathrm{~m}$. Then there are $8 \mathrm{~m} b_{i}$'s in the interval $(10 \mathrm{~m}, 30 \mathrm{~m})$, and the quotient of any two of them is less than 3, so none of them can divide any other. Thus the answer is 8 m.