Video: Common Vertices Problem Using Roots of Unity

Two regular polygons are inscribed in the same circle: the first has 1731 sides and the second has 4039. If the two polygons have at least one vertex in common, how many vertices in total will coincide?

01:49

Video Transcript

Two regular polygons are inscribed in the same circle. The first has 1731 sides. And the second has 4039. If the two polygons have at least one vertex in common, how many vertices in total will coincide?

Remember, the geometrical interpretation of the 𝑛th roots of unity on an Argand diagram is as the vertices of a regular 𝑛-gon inscribed within a unit circle whose centre is the origin. This means then that we can say that, to solve this problem, we need to find the number of common roots of 𝑧 to the power of 1731 minus one equals zero and 𝑧 to the power of 4039 minus one equals zero. Remember, the common roots of 𝑧 to the power of π‘š minus one equals zero and 𝑧 to the power of 𝑛 minus one equals zero are the roots of 𝑧 to the power of 𝑑 minus one equals zero, where 𝑑 is the greatest common divisor of π‘š and 𝑛.

So we know that the common roots of our two equations are the roots of 𝑧 to the power of 𝑑 minus one equals zero, where 𝑑 is the greatest common divisor of 1731 and 4039. And this means if we can find the value of 𝑑, the greatest common divisor of 1731 and 4039, that will tell us how many common roots there actually are. As a product of their prime factors, they can be written as three times 577 and seven times 577, respectively. So their greatest common divisor and the value of 𝑑 is 577. And this means that as long as the polygons have one vertex in common, they will actually have a total of 577 vertices that coincide.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.