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

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by Max@Math Revolution » Thu Oct 31, 2019 12:03 am
[GMAT math practice 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
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by GMATGuruNY » Thu Oct 31, 2019 3:37 am
Max@Math Revolution wrote:[GMAT math practice 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
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 = [(n-2)(180)]/n = (180n - 360)/n = 180 - 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

The correct answer is B.
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by Max@Math Revolution » Sun Nov 03, 2019 6:18 pm
=>

Since the sum of all interior angles of an n-polygon is 180(n-2), an interior angle of an n-regular polygon = 180(n-2)/n = (180n - 360)/n = 180 - (360/n).
In order for 180 - (360/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.
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