An inner angle of an n-regular polygon is a positive integer. How many

Question

An inner angle of an n-regular polygon is a positive integer. How many possible n’s are there less than or equal to 20?

A. 10 
B. 11 
C. 12 
D. 13 
E. 14

chetan2u · Answer

The sum of angles in a n-sided polygon is (n-2)*180, and so each inner angle = \frac{(n-2)*180}{n} 
Now  \frac{(n-2)*180}{n}  will be an integer
(1)  When n is a factor of 180.. .180=2*2*3*3*5
So n can be 3, 4, 5, 6, 9, 10, 12, 15, 18 and 20......Minimum is 3 as polygon has to have at least 3 angles---11 values
(2)  Apart from that we can check for the remaining number if n is a factor of (n-2)*180.. 
n=7......(7-2)*180=5*180....NO
n=8......(8-2)*180=6*180=8*27*5.....YES
n=11.....9*180...NO
n=13....11*180...NO
n=14....12*180..NO
Similarly 16, 17 and 19 will also give a NO as the answer

Total -10+1=11

B
------------------------------------------------------------------
 Aderonke01 
There are 10 from (I) and 1 from (II), so total is 10+1=11..

Another straight forward method would be to simplify    \frac{(n-2)*180}{n}=180-\frac{360}{n} 
Now  180-\frac{360}{n}  will be an integer, and for that 360/n should be an integer..
360/n will be an integer when n is a factor of  360=2^3*9*5 ....
factors  \leq{20}  :   3, 4, 5, 6, 8, 9, 10, 12, 15, 18 and 20  -----11 values

Aderonke01 · Answer

Hi, from the first method: the total is 10: So n can be 3, 4, 5, 6, 9, 10, 12, 15, 18 and 20... How come your count is 11?

Posted from my mobile device

GMATGuruNY · Answer

Sum of the angles of an n-sided polygon = (n-2)(180), where n ≥ 3.

In a regular n-sided polygon, the degree measurement of each interior angle is THE SAME.
Since there are n angles in total, we get:
Each angle  = \frac{(n-2)(180)}{n} = \frac{180n - 360}{n} = 180 - \frac{360}{n} .

For the degree measurement to be an integer, n must be a factor of 360.
Factors of 360 between 3 and 20, inclusive:
3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20 --> 11 options

B

EgmatQuantExpert · Answer

Solution

Given  
 • An inner angle of an n-regular polygon is a positive integer.

To find  
 • The number of possible value of n that are less than or equal to 20

Approach and Working out  
 • Internal angle of an n-sided regular polygon = 180- 360/n
• = 180(n-2)/2
• Since internal angle is a positive integer, 180 or 180 (n-2) should be a multiple of n.

Case 1- 180 is a multiple of n 
 • Factors of 180 = 1, 2, 3, 4, 5, 6, 9,10,12, 15,18, and 20 
However, n cannot be 1 or 2 as 1 side and 2 side will never form a polygon.

Case 2 – 180* (n-2) is a multiple of n  
 • Let’s check for n=7, 8, 11, 13, 14, 16, 17, and 19
• n=7 => 180 *5=> Not divisible by 7
• n= 8=> 180 * 6 => Divisible by 8
• n= 11=> 180*9/11=> Not divisible by 11
• n = 13 => 180 *11/13 => Not divisible by 13
• n= 16 => 180*14/16 => Not divisible by 16
• n= 17 => 180*15/17 => not divisible by 17
• n = 19 = 180*17/19 => not divisible by 19

Hence, there are 11 values of n: 3, 4, 5, 6, 8, 9,10,12, 15,18, and 20

Thus, option B is the correct answer.
  Correct Answer: Option B

MathRevolution · Answer

=>

Since the sum of all interior angles of an n-polygon is  180(n-2) , an interior angle of an n-regular polygon =  180^o\frac{(n-2)}{n} = (\frac{180^on – 360^o)}{n} = 180^o – (\frac{360^o}{n}).

In order for  180^o - (\frac{360^o}{n})  to be an integer,  n  must be a factor of  360 .

Then the possible values of  n  are  3, 4, 5, 6, 8, 9, 10, 12, 15, 18,  and  20.

We have  11  possible values of  n.

Therefore, B is the answer.

KarishmaB · Answer

To avoid manual calculation and hence chances of missing out some numbers, we can use factors and factorisation here. The interior angle of an n sided regular polygon is given by \frac{180(n-2)}{n} = 180 - \frac{360}{n} For this to be an integer, we need all values of n (greater than 2 to form a polygon) that are factors of 360. 360 = 2^3 * 3^2 * 5 Total number of factors = 4*3*2 = 24 Since square root of 360 is between 18 and 19, 12 factors will be 18 and below while 12 factors will be 19 and above. 18 is a factor of 360 and right after it we have 20. Hence, 360 has 13 factors 20 or below. Of those, two are 1 and 2 so we ignore them and are left with 11 factors. Hence n can take 11 values. Answer (B) Check these videos for factors and factorisation: Factors: https://youtu.be/DxIH8rjhpKY Factorisation: https://youtu.be/Kd-4cH4cqHw and this post: https://anaprep.com/number-properties-f ... -a-number/

An inner angle of an n-regular polygon is a positive integer. How many

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An inner angle of an n-regular polygon is a positive integer. How many

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