For integers $L$ and $M$ greater than $1,$ prove that the following sets are equivalent if and only if $L$ and $M$ are coprime.
$$\bigg\{\large e^{\big(\tfrac{-i\text{ }2\pi \text{ }k}{L}\big)}\bigg\}_{k = 0}^{k = L-1}\text{ } \large=\text{ } \bigg\{\large e^{\big(\tfrac{-i\text{ }2\pi \text{ } k \text{ }M}{L}\big)}\bigg\}_{k = 0}^{k = L-1} \iff L \perp M$$