28.23 A sample consisting of 5 positive integers has an arithmetic mean of 10 and a median of 12. The smallest possible difference between the largest and smallest numbers in the sample is
(A) 2.
(B) 3.
(C) 5.
(D) 7.
(E) 10.
(45th American High School Mathematics Examination, 1994)
[Solution] Let the five numbers be $a \leqslant b \leqslant c \leqslant d \leqslant e$. By the problem, $c=12$, and
$$
a+b+c+d+e=10 \times 5=50.
$$

Also, $d \geqslant 12, e \geqslant 12$, so $c+d+e \geqslant 3c=36$, $\therefore 2a \leqslant a+b \leqslant 50-36=14$, which means $a \leqslant 7$.
Thus, $e-a \geqslant 5$, with equality holding only when $a=b=7, c=d=e=12$. Therefore, the answer is $(C)$.