1993 AMC 12 第 15 题

Question

有多少个

n

值能使正

n

边形的内角度数为整数？

For how many values of

n

will an

n

-sided regular polygon have interior angles with integral degree measures?

16

18

20

22

24

一个内角为

180^\circ-\frac{360^\circ}{n}

。

An interior angle is

180^\circ-\frac{360^\circ}{n}

数出因数

n

的所有可能值，使其整除

360

，并排除小于

3

的值。

Count divisors

n

of

360

and exclude values below

3

内角为

180^\circ-\frac{360^\circ}{n}

，其度数为整数当且仅当

360

能被

n

整除。因为

360=2^3\cdot3^2\cdot5

，它有

(3+1)(2+1)(1+1)=24

个正因数。排除

n=1

和

n=2

后，剩下

22

个可行值。因此正确答案是 D。

The interior angle is

180^\circ-\frac{360^\circ}{n},

which is integral exactly when

360

is divisible by

n.

Since

360=2^3\cdot3^2\cdot5,

it has

(3+1)(2+1)(1+1)=24

positive divisors. Excluding

n=1

and

n=2

leaves

22

allowable values. Thus the correct answer is D.

其他年份的第 15 题