1993 AMC 12 Problem 15

Attempt Problem 15 of the 1993 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1993 AMC 12 solutions, or check the answer key.

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15.

For how many values of

n

will an

n

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

16

18

20

22

24

An interior angle is

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

Count divisors

n

360

and exclude values below

3

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.

Problem 15 in other years