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Difference between revisions of "1993 AHSME Problems/Problem 15"

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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
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\text{(E) } 24\quad
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\text{(E) } 24</math>
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\text{(F) } 69</math>
 
  
 
== Solution ==
 
== Solution ==
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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>.
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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>.
  
 
== See also ==
 
== See also ==

Latest revision as of 12:39, 6 November 2020

Problem

For how many values of $n$ will an $n$-sided regular polygon have interior angles with integral measures?

$\text{(A) } 16\quad \text{(B) } 18\quad \text{(C) } 20\quad \text{(D) } 22\quad \text{(E) } 24$

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 $2^3 * 3^2 * 5$. 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 $\fbox{D}$.

See also

1993 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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