$29 \cdot 12$ If the three-digit number $\overline{2 a 3}$ plus 326 results in another three-digit number $\overline{5 b 9}$, and if $\overline{569}$ is divisible by 9, then $a+b$ equals
(A) 2.
(B) 4.
(C) 6.
(D) 8.
(E) 9.
(18th American High School Mathematics Examination, 1967)
[Solution] Given $\overline{5 b 9}$ is divisible by 9 $(0 \leqslant b \leqslant 9)$, then
$$
\frac{\overline{5 b 9}}{9}=10 \cdot \frac{50+b}{9}+1
$$

is an integer.
That is, $50+b$ is divisible by 9, and since $0 \leqslant b \leqslant 9$, then $b=4$.
Also, $\overline{2 a 3}=\overline{5 b 9}-326=549-326=223$,
thus $a=2$, and $a+b=6$. Therefore, the answer is $(C)$.