$19 \cdot 27$ The sum of the $n$ interior angles and one exterior angle of a convex $n$-sided polygon is $1350^{\circ}$, then $n$ equals
(A) 6.
(B) 7.
(C) 8.
(D) 9.
(E) 10.
(China Beijing Junior High School Mathematics Competition, 1984)
[Solution] The sum of the interior angles of a convex $n$-sided polygon is $(n-2) \cdot 180^{\circ}$. Let one of the exterior angles be $\alpha$, then
$$
(n-2) \cdot 180^{\circ} + \alpha = 1350^{\circ} \text{, and } 0^{\circ} < \alpha < 180^{\circ} \text{. }
$$

By the property of integer divisibility, $90 \mid \alpha$, so $\alpha = 90^{\circ}$, and solving for $n$ gives $n = 9$.
Therefore, the answer is $(D)$.