If $a_{1} b_{1} c_{1}, a_{2} b_{2} c_{2}$ and $a_{3} b_{3} c_{3}$ are three-digit numbers, each of which is divisible by a non zero integer k, then $∆ = |\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}|$ is always

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divisible by k

divisible by $k^{2}$

divisible by 2k

not divisible by k

answer is A.

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Detailed Solution

Since $a_{1} b_{1} c_{1}, a_{2} b_{2} c_{2}$ and $a_{3} b_{3} c_{3}$ are divisible by k, therefore

$\begin{matrix} 100 a_{1} + 10 b_{1} + c_{1} = n_{1} k \\ 100 a_{2} + 10 b_{2} + c_{2} = n_{2} k \\ 100 a_{3} + 10 b_{3} + c_{3} = n_{3} k \end{matrix}$

where n1, n2, n3 are integers.

Now $Δ = |\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}| = |\begin{matrix} a_{1} & b_{1} & 100 a_{1} + 10 b_{1} + c_{1} \\ a_{2} & b_{2} & 100 a_{2} + 10 b_{2} + c_{2} \\ a_{3} & b_{3} & 100 a_{3} + 10 b_{3} + c_{3} \end{matrix}|$

(Applying $C_{3} \to C_{3} + 10 C_{2} + 100 C_{1})$

$= |\begin{matrix} a_{1} & b_{1} & n_{1} k \\ a_{2} & b_{2} & n_{2} k \\ a_{3} & b_{3} & n_{3} k \end{matrix}| = K |\begin{matrix} a_{1} & b_{1} & n_{1} \\ a_{2} & b_{2} & n_{2} \\ a_{3} & b_{3} & n_{3} \end{matrix}| = k Δ_{1}$