### Instruction: For any integer $n>1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is:
$\text{(A) } 0\quad\qquad \text{(B) } 1\quad\\ \text{(C) } \frac{n}{2} \text{ for n even, } \frac{n+1}{2} \text{ for n odd}\quad\\ \text{(D) } n-1\quad \text{(E) } n$.### Response: I know that $n!$ is the product of all positive integers from 1 to $n$, so it has many factors.
Therefore, any number of the form $n!+k$ where $k$ is a positive integer less than or equal to $n$ must also have a factor in common with $n!$, namely $k$.
That means $n!+k$ is not prime for any $k \leq n$, so there are no prime numbers between $n!+1$ and $n!+n$.