Given that $a, b, c$ are integers such that $a b c = 60$ and $ω$ is a non real cube root of unity. Then the minimum possible value of $| a + b ω + c ω^{2} |$ is

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$\sqrt{3}$

answer is B.

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Detailed Solution

${| a + b ω + c ω^{2} |}^{2} = (a + b ω + c ω^{2}) (a + b ω^{2} + c ω) = a^{2} + b^{2} + c^{2} - a b - b c - c a = \frac{1}{2} ({(a - b)}^{2} + {(b - c)}^{2} + {(c - a)}^{2})$
Now to minimize, choose $a = 3, b = 4, c = 5.$

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