$30 \cdot 23$ Given a sequence with 58 terms, each term is of the form $p+n$, where $p$ represents the product of all prime numbers less than or equal to 61, i.e., $2,3,5, \cdots, 61$, and $n$ takes the values $2,3,4, \cdots, 59$ in sequence. Let $N$ be the number of primes that appear in this sequence, then $N$ is
(A) 0.
(B) 16.
(C) 17.
(D) 57.
(E) 58.
(11th American High School Mathematics Examination, 1960)
［Solution］Notice the following expressions:
$$
\begin{array}{l}
2 \cdot 3 \cdot 5 \cdot \cdots \cdot 61+2=2(3 \cdot 5 \cdot \cdots \cdot 61+1), \\
2 \cdot 3 \cdot 5 \cdot \cdots \cdot 61+3=3(2 \cdot 5 \cdot \cdots \cdot 61+1), \\
2 \cdot 3 \cdot 5 \cdot \cdots \cdot 61+4=2(3 \cdot 5 \cdots \cdots \cdot 61+2), \\
\cdots \cdots \cdots \cdots \cdots \cdots
\end{array}
$$

That is, when $n$ takes the values $2,3,4, \cdots, 59$, $p+n$ always has a prime factor, so $p+n$ is always composite, or $N=0$.
Therefore, the answer is $(A)$.