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 For how many values of <math>n</math> will an <math>n</math>sided regular polygon have interior angles with integral measures?   For how many values of <math>n</math> will an <math>n</math>sided regular polygon have interior angles with integral measures? 
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−  POLYGONS ARE BOOGERS
 
   
 <math>\text{(A) } 16\quad   <math>\text{(A) } 16\quad 
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 \text{(C) } 20\quad   \text{(C) } 20\quad 
 \text{(D) } 22\quad   \text{(D) } 22\quad 
−  \text{(E) } 24\quad  +  \text{(E) } 24</math> 
−  \text{(F) } 69</math>
 
   
 == Solution ==   == Solution == 
−  Start with the facts that all polygons have they're 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 <math>2^3 * 3^2 * 5</math>. 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 <math>\fbox{D}</math>.  +  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 <math>2^3 * 3^2 * 5</math>. 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 <math>\fbox{D}</math>. 
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−  THERE IS NO SOLUTION TO THIS PROBLEM
 
−  NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN NEIN 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 == See also ==   == See also == 
Latest revision as of 13:39, 6 November 2020
Problem
For how many values of will an sided regular polygon have interior angles with integral measures?
Solution
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 . 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 .
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.