1. Let $p$ be a prime number greater than 3, and $p+8$ is also a prime number. Let $S_{1}=p(p+8)$, and $S_{i+1} (i=1,2, \cdots, k-1)$ denote the sum of the digits of $S_{i}$. If $S_{k}$ is a single-digit number, for all possible prime numbers $p$, $S_{k}$ has ( ) different values.
(A) 1
(B) 2
(C) 3
(D) 4
$-1 . A$.
Since $p$ is a prime number greater than 3, and $p+8$ is also a prime number, therefore, $p$ is a prime of the form $3M+2$. So,
$$
p(p+8)=(p+4)^{2}-16 \equiv 2(\bmod 9) \text {. }
$$

Also, $S_{i+1} \equiv S_{i}(\bmod 9)$, then $S_{k}=2$, which means $S_{k}$ has only one value.