Subgroup of Circle Group Generated by Distinct Roots of Unity

Theorem
Let $K$ be the circle group.

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

Let $d = \gcd \set {m, n}$ be the greatest common divisor of $m$ and $n$.

Let $\alpha$ be a primitive $n$th root of unity.

Let $\beta$ be a primitive $m$th root of unity.

Let $\gamma$ be a primitive $d$th root of unity.

Let $H = \gen {\alpha, \beta}$ be the subgroup of $K$ generated by $\alpha, \beta$.

Then $H = \gen \gamma$.