Products Of Primitive Roots Of Unity

Algebra Level 5

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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