$$\begin{gathered} \mathrm{Let} \mathrm{the} \mathrm{three} \mathrm{digit} \mathrm{numbers} \mathrm{A}28, 3\mathrm{B}9, 62\mathrm{C} \\ \mathrm{where} \mathrm{A}, \mathrm{B}, \mathrm{C} \mathrm{are} \mathrm{integers} \mathrm{between} 0 \mathrm{and} 9 \mathrm{be} \\ \mathrm{divisible} \mathrm{by} \mathrm{a} \mathrm{fixed} \mathrm{integer} \mathrm{K} \mathrm{then} \\ \left|\begin{array}{ccc}\mathrm{A}& 3& 6\\ 8& 9& \mathrm{C}\\ 2& \mathrm{B}& 2\end{array}\right| \mathrm{is} \mathrm{divisible} \mathrm{by} \end{gathered}$$

Let $\mathrm{A}28=\mathrm{ka}, 3\mathrm{B}9=\mathrm{kb}, 62\mathrm{C}=\mathrm{kc}$ be divisible by 'k' when a,b,c an integers.

      apply ${\mathrm{R}}_2\to {\mathrm{R}}_2+100{\mathrm{R}}_1+10{\mathrm{R}}_3$

$\left|\begin{array}{ccc}\mathrm{A}& 3& 6\\ \mathrm{A}28& 3\mathrm{B}9& 62\mathrm{C}\\ 2& \mathrm{B}& 2\end{array}\right|=\mathrm{k}\left|\begin{array}{ccc}\mathrm{A}& 3& 6\\ \mathrm{a}& \mathrm{b}& \mathrm{c}\\ 2& \mathrm{B}& 2\end{array}\right|$

       $=$ integral multiple of k

        $\therefore ∆$ is divisible by 'k'