$15 \cdot 13$ In the subset $S$ of the set $\{1,2,3, \cdots, 50\}$, the sum of any two elements cannot be divisible by 7. The maximum number of elements in such a subset $S$ is
(A) 6.
(B) 7.
(C) 14.
(D) 22.
(E) 23.
(43rd American High School Mathematics Examination, 1992)
[Solution] For natural numbers $m=7 k_{1}+r_{1}, \quad n=7 k_{2}+r_{2}$, if $r_{1}+r_{2}$ cannot be divided by 7, then $m+n$ cannot be divided by 7 either.
Thus, the subset $S$ with the maximum number of elements is
$$
\{x \mid x=7 k+r, r=1,2,3 \text {, and } x \leqslant 50\} \cup\{7\},
$$

which contains 23 elements.
Therefore, the answer is $(E)$.