# Products Of Primitive Roots Of Unity

Algebra Level 4

Let $\zeta_m$ be a primitive $m^\text{th}$ root of unity, and
let $\zeta_n$ be a primitive $n^\text{th}$ root of unity.
Then $\zeta_m\zeta_n$ is a primitive $\ell^\text{th}$ root of unity for some positive integer $\ell.$

What can we say about $\ell$ in general?


Clarification: In the answer choices, $\gcd(\cdot)$ and $\text{lcm}(\cdot)$ denote the greatest common divisor function and the lowest common multiple function, respectively.

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