$~~~~~~~~~~~$

It is easy to check that $2$ is not in $S$. For all other positive integers $n$, either $n$ and $n + 2$ or $n + 7$ and $n + 9$ are even and composite, ergo we will have at least $7$ consecutive positive integers. To show that $7$ is the maximum, observe the set {$n, n + 1, ..., n + 9$} for $n = 90$.

can u explain it in details?

Lukaluce wrote: Let $S$ be the set of all positive integers $n$ such that each of the numbers $n + 1$, $n + 3$, $n + 4$, $n + 5$, $n + 6$, and $n + 8$ is composite. Determine the largest integer $k$ with the following property: For each $n \in S$ there exist at least $k$ consecutive composite integers in the set {$n, n +1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9$}. Can someone kindly explain me this question ? I don't understand this question:(