elementary number theory - How do I prove that for every positive integer $n$, there exist $n$ consecutive positive integers, each of which is composite?

I need help proving that for every positive integer $n$, there exist $n$ consecutive positive integers, each of which is composite. The hint that came with the problem is: Consider the numbers $2+(n+1)!,3+(n+1)!,...,n+(n+1)!,n+1+(n+1)!$.

I honestly have no idea how to begin here. I am specifically confused about what it means when it says "there exist $n$ consecutive positive integers". Does this mean that if $n = 5$, for example, then somewhere in the positive integers there are 5 consecutive composite integers, and that we want to prove that? I get that composite means that they are not prime, so they have more factors than 1 and themselves.

Any help would be greatly appreciated.

Thank you!

Answer

The numbers you are given are consecutive, right? Are they composite? Hint: For the term $i + (n+1)!$, try to factor out $i$...