Two polygons in a circle

Call the polygon with number of sides $2016$ $A$ and call the other polygon $B$ .

In the argand plane, sides of a $A$ are represented by the $2016th$ roots of unity and sides of $B$ are represented by the $nth$ roots of unity.

Thus the number of common vertices is given by the number of common roots of $z^{2016}-1=0$ and $z^{n}-1=0$ which implies number of common vertices $=gcd(2016,n)$ .

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 $2016$ which are co-prime with it.

Thus our answer is $\phi (2016) - 1 = 575$ . $\text{Since n is not 1.}$

$\phi (x)$ represents the Euler Totient Function .

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Generalization:

For two regular polygons with number of sides $m$ and $n$ ,number of common vertices is $\gcd(m,n)$ .