$\begin{array}{1 1}(A)\;min(p,q)&(B)\;max(p,q)\\(C)\;zero&(D)\;1\end{array} $

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$p^{th}$ root of unity lies at an angle difference of $\large\frac{2\pi}{p}$ on a unit circle in argand plane.

Similarly $q^{th}$ root of unity lies on a difference of $\large\frac{2\pi}{q}$ angle on unit circle.

Since P and q are prime therefore $\large\frac{2\pi k}{p}$ and $\large\frac{2\pi m}{q}$ are not same for any +ve integer k and m.

Thus only common $p^{th}$ and $q^{th}$ root is 1 which is not imaginary.Thus no common imaginary root is possible.

Hence (C) is the correct answer.

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