$32 \cdot 45$ For any integer greater than 1, the number of primes greater than $n!+1$ and less than $n!+n$ is
(A) 0.
(B) 1.
(C) $\frac{n}{2}$, when $n$ is even, $\frac{n+1}{2}$, when $n$ is odd.
(D) $n-1$.
(E) $n$.
(20th American High School Mathematics Examination, 1969)
[Solution] Since $k$  |  $(n!+k), \quad k=2,3, \cdots, n-1$. Therefore, all integers between $n!+1$ and $n!+n$ (excluding the endpoints) are composite. Hence, the answer is $(A)$.