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Olympiad workbook
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Olympiad workbook
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IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Let X be the set of all even positive integers n such that the...

Let XX be the set of all even positive integers nn such that the measure of the angle of some regular polygon is nn degrees. Find the number of elements in XX.

Correct answer

16

Step-by-step solution

Let number of sides of polygon be PP. ∴(P−2)⋅180∘P=n∘\therefore \frac{(P-2) \cdot 180^{\circ}}{P}=n^{\circ} then P(180−n)=360P(180-n)=360 ∴P=360180−n\therefore P=\frac{360}{180-n} But n=2kn=2 k then P=18090−kP=\frac{180}{90-k} Hence possible values of (90−k)(90-k) are 1,2,3,4,5,6,9,10,12,15,18,20,30,36,451,2,3,4,5,6,9,10,12,15,18,20,30,36,45 and 60 ∴16\therefore 16 such polygons are possible.
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