Intersection of Multiplicative Groups of Complex Roots of Unity

Theorem
Let $\struct {K, \times}$ denote the circle group.

Let $m, n \in \Z_{>0}$ be (strictly) positive integers.

Let $c = \lcm \set {m, n}$ be the lowest common multiple of $m$ and $n$.

Let $\struct {U_n, \times}$ denote the multiplicative group of complex $n$th roots of unity.

Let $\struct {U_m, \times}$ denote the multiplicative group of complex $m$th roots of unity.

Let $H = U_m \cap U_n$.

Then $H = U_c$.