For all $n>1$, find $n$ consecutive non-prime integers.
Fix an integer $N$ and let's see what we can say about the divisors of the consecutive integers following $N$.

![](https://cdn.mathpix.com/cropped/2024_05_10_8c44cd8b5c585c15e58bg-345.jpg?height=71&width=771&top_left_y=1889&top_left_x=488)

We know nothing about $N+1$, except that it is coprime with $N$. However, if $N$ is even, we can be sure that $N+2$ is even (as are all numbers of the form $N+2i$). If $N$ is a multiple of 3, then $N+3$ will necessarily be divisible by 3. More generally, for any integer $k$, if $N$ is divisible by $k$, then $N+k$ is also divisible by $k$. To have $n$ consecutive non-prime integers, we need to impose conditions on $N$ so as to provide divisors for the $n$ consecutive numbers $N+2, N+3, \ldots, N+(n+1)$. It is sufficient that $N$ is divisible by all the numbers from 2 to $n+1$; by choosing $N=(n+1)!$, we are done.