Two regular polygons are inscribed in the same circle. First polygon has 2016 sides and the second polygon has sides.
If the polygons have only one vertex in common, find the number of possible values of .
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Call the polygon with number of sides 2 0 1 6 A and call the other polygon B .
In the argand plane, sides of a A are represented by the 2 0 1 6 t h roots of unity and sides of B are represented by the n t h roots of unity.
Thus the number of common vertices is given by the number of common roots of z 2 0 1 6 − 1 = 0 and z n − 1 = 0 which implies number of common vertices = g c d ( 2 0 1 6 , n ) .
It is given in the question that this value is one.
Therefore, number of possible values of n is all the integers less than or equal 2 0 1 6 which are co-prime with it.
Thus our answer is ϕ ( 2 0 1 6 ) − 1 = 5 7 5 . Since n is not 1.
ϕ ( x ) represents the Euler Totient Function .
Generalization:
For two regular polygons with number of sides m and n ,number of common vertices is g cd ( m , n ) .