https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=X-bob-x&feedformat=atom AoPS Wiki - User contributions [en] 2022-05-19T01:49:42Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=1993_AHSME_Problems/Problem_15&diff=136748 1993 AHSME Problems/Problem 15 2020-11-06T17:39:01Z <p>X-bob-x: /* Solution */</p> <hr /> <div>== Problem ==<br /> <br /> For how many values of &lt;math&gt;n&lt;/math&gt; will an &lt;math&gt;n&lt;/math&gt;-sided regular polygon have interior angles with integral measures?<br /> <br /> &lt;math&gt;\text{(A) } 16\quad<br /> \text{(B) } 18\quad<br /> \text{(C) } 20\quad<br /> \text{(D) } 22\quad<br /> \text{(E) } 24&lt;/math&gt;<br /> <br /> == Solution ==<br /> Start with the facts that all polygons have their exterior angles sum to 360 and the exterior and interior angles make a linear pair of angles. So our goal is to find the number of divisors of 360 to make both the interior and exterior angles integers. The prime factorization of 360 is &lt;math&gt;2^3 * 3^2 * 5&lt;/math&gt;. That means the number of divisors is 4*3*2 = 24. But we're not done yet. We cannot have a 1 or 2 sided polygon so we subtract off two bringing us to our final answer of 22 &lt;math&gt;\fbox{D}&lt;/math&gt;.<br /> <br /> == See also ==<br /> {{AHSME box|year=1993|num-b=14|num-a=16}} <br /> <br /> [[Category:Introductory Geometry Problems]]<br /> {{MAA Notice}}</div> X-bob-x