### 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 πth roots of unity on an Argand diagram is as the vertices of
a regular π-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 π§ to the
power of 1731 minus one equals zero and π§ to the power of 4039 minus one equals
zero. Remember, the common roots of π§ to
the power of π minus one equals zero and π§ to the power of π minus one equals
zero are the roots of π§ to the power of π minus one equals zero, where π is the
greatest common divisor of π and π.

So we know that the common roots of
our two equations are the roots of π§ to the power of π minus one equals zero,
where π is the greatest common divisor of 1731 and 4039. And this means if we can find the
value of π, 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 π 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.