$32 \cdot 3$ If an integer is a two-digit number, equal to $k$ times the sum of its digits, and if the digits are interchanged, the new number is 
(A) $(9-k)$ times.
(B) $(10-k)$ times.
(C) $(11-k)$ times.
(D) $(k-1)$ times.
(E) $(k+1)$ times.
(3rd American High School Mathematics Examination, 1952)
［Solution］Let the tens digit of the integer be $u$, and the units digit be $v$. According to the problem, we have
$$
\left\{\begin{array}{l}
10 u+v=k(u+v) \\
10 v+u=x(u+v) .
\end{array}\right.
$$

Here, $x$ is the multiple of the sum of the digits after the digits are swapped.
From the above two equations, we can solve that $x=11-k$.
Therefore, the correct choice is $(C)$.