$10 \cdot 22$ The sum of the first $n$ terms of an arithmetic sequence is 153, the common difference is 2, the first term is an integer, and $n>1$, then the number of possible values for $n$ is
(A) 2.
(B) 3.
(C) 4.
(D) 5.
(E) 6.
(15th American High School Mathematics Examination, 1964)
［Solution］Let $a$ be the first term of the arithmetic sequence. According to the problem, we have
$$
\frac{n}{2}\{a+[a+2(n-1)]\}=153 \text {, }
$$

or
$$
n^{2}+(a-1) n-153=0 \text {. }
$$

The product of the roots of this quadratic equation is -153, so the possible values of $n$ among the factors greater than 1 are: $3,9,17,51,153$.
Therefore, the answer is $(D)$.