If |111abcbca|=2023 then the number of ways in which |ab2bc2ac3c−3a23a−3b3b−3cc−b2a−2c2b−2a|+|3c23a23b22c−2a2a−2b2b−2cb−cc−aa−b| can be expressed as product of 3 relatively prime positive integers is

If |111abcbca|=2023 then the number of ways in which |ab2bc2ac3c−3a23a−3b3b−3cc−b2a−2c2b−2a|+|3c23a23b22c−2a2a−2b2b−2cb−cc−aa−b| can be expressed as product of 3 relatively prime positive integers is

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If $| \begin{matrix} 1 & 1 & 1 \\ a & b & c \\ b & c & a \end{matrix} | = 2023$ then the number of ways in which $| \begin{matrix} a b & 2 b c & 2 a c \\ \frac{3 c - 3 a}{2} & 3 a - 3 b & 3 b - 3 c \\ c - b & 2 a - 2 c & 2 b - 2 a \end{matrix} | + | \begin{matrix} 3 c^{2} & 3 a^{2} & 3 b^{2} \\ 2 c - 2 a & 2 a - 2 b & 2 b - 2 c \\ b - c & c - a & a - b \end{matrix} |$ can be expressed as product of 3 relatively prime positive integers is

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$11$

$14$

$17$

$20$

answer is B.

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Detailed Solution

$| \begin{matrix} a b & 2 b c & 2 a c \\ \frac{3 c - 3 a}{2} & 3 a - 3 b & 3 b - 3 c \\ c - b & 2 a - 2 c & 2 b - 2 a \end{matrix} | + | \begin{matrix} 3 c^{2} & 3 a^{2} & 3 b^{2} \\ 2 c - 2 a & 2 a - 2 b & 2 b - 2 c \\ b - c & c - a & a - b \end{matrix} | = | \begin{matrix} a b & b c & a c \\ 3 c - 3 a & 3 a - 3 b & 3 b - 3 c \\ 2 c - 2 b & 2 a - 2 c & 2 b - 2 a \end{matrix} | + | \begin{matrix} 3 c^{2} & 3 a^{2} & 3 b^{2} \\ 2 c - 2 a & 2 a - 2 b & 2 b - 2 c \\ b - c & c - a & a - b \end{matrix} | = 6 | \begin{matrix} a b - c^{2} & b c - a^{2} & a c - b^{2} \\ c - a & a - b & b - c \\ c - b & a - c & b - a \end{matrix} | = 2 \times 3 \times 7^{2} \times 17^{4}$