13. If the ratio of the degree measures of the $n$ interior angles of a convex $n$-sided polygon is 1:2:3: $\ldots: n$, then the number of possible values for $n$ is ( )
(A) 1
(B) 2
(C) 3
(D) infinitely many
Answer: (B)
Solution: Let the $n$ interior angles of the deformed $n$-sided polygon be $x^{\circ}, 2 x^{\circ}, 3 x^{\circ}, \ldots, n x^{\circ}$, then by the formula for the sum of interior angles of a polygon, we get
$$
x+2 x+3+\cdots+n x=(n-2) 180,
$$

which is
$$
\frac{n(n+1) x}{n-2}=360
$$

If $n=3$, then
$$
x=30,
$$

the three interior angles of the triangle are
$$
30^{\circ}, 60^{\circ}, 90^{\circ} \text {, }
$$

so $n$ can be 3.
If $n=4$, then
the four interior angles of the quadrilateral are
$$
\begin{array}{c}
x=36, \\
36^{\circ}, 72^{\circ}, 108^{\circ}, 144^{\circ},
\end{array}
$$

so $n$ can be 4.

If $n \geq 5$, then
$$
n x=360 \frac{n-2}{n+1} \geq 180 \text { . }
$$

Therefore, the possible values of $n$ are only 2.
Hence, the answer is (B).