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Let the three digit numbers A34, 5B1, 72C where A, B, C are integers between 0 and 9 be divisible by a fixed integer K then $| \begin{matrix} A & 5 & 7 \\ 4 & 1 & C \\ 3 & B & 2 \end{matrix} |$ is divisible by

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$K^{2}$

$K + 2$

$K + 1$

answer is A.

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Detailed Solution

${| \begin{matrix} A & 5 & 7 \\ 4 & 1 & C \\ 3 & B & 2 \end{matrix} |}^{100 R_{1} + 10 R_{3} + R_{2}} = | \begin{matrix} A 34 & 5 B 1 & 72 C \\ 4 & 1 & C \\ 3 & B & 2 \end{matrix} |$

$= | \begin{matrix} K . n_{1} & K . n_{2} & K . n_{3} \\ 4 & 1 & C \\ 3 & B & 2 \end{matrix} |$ where $n_{1}, n_{2}, n_{3} \in Ζ$

$= K | \begin{matrix} n_{1} & n_{2} & n_{3} \\ 4 & 1 & C \\ 3 & B & 2 \end{matrix} |$ is divisible by K

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