Given the numbers $1,2,3, \ldots, 1000$. Find the largest number $m$, such that: no matter which $m$ of these numbers are erased, among the remaining $1000-m$ numbers, there will be two such that one divides the other.
If $m \geq 500$, then, by erasing the first $m$ numbers (from 1 to $m$), we will leave the numbers from $m+1 \geq 501$ to 1000, among which none, obviously, divides another (all pairwise ratios are less than two).

The number $m=499$ has the required property. Indeed, if we take 501 numbers from the set of 1 to 1000, then among them there will be two, one of which divides the other (see problem $\underline{60299}$ ).

## Answer

$m=499$.