$12 \cdot 1$ If $S=i^{n}+i^{-n}$, where $i$ is the imaginary unit, and $n$ is an integer, then the number of distinct possible values of $S$ is
(A) 1 .
(B) 2 .
(C) 3 .
(D) 4 .
(E) more than 4 .
(8th American High School Mathematics Examination, 1957)
［Solution］Consider four cases:
(1) When $n=4 k$, $\quad S=i^{4 k}+i^{-4 k}=2$;
(2) When $n=4 k+1$, $S=i^{4 k+1}+i^{-(4 k+1)}=0$;
(3) When $n=4 k+2$, $\quad S=i^{4 k+2}+i^{-(4 k+2)}=-2$;
(4) When $n=4 k+3$, $S=i^{4 k+3}+i^{-(4 k+3)}=0$. Therefore, the answer is $(C)$.