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If p and q are distinct prime numbers then the no of distinct imaginary numbers which are $p^{th}$ as well as $q^{th}$ root of unity are

$\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.
answered Apr 10, 2014 by sreemathi.v

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