Question

If $abc,lmn$ and $pqr$ be any three-digit numbers each of which is divisible by $k$, then show that
$\Delta =\left|\begin{array}{ccc}a& b& c\\ l& m& n\\ p& q& r\end{array}\right|$ is also divisible by $k.$

Answer 1

$abc=100a+10b+c=n_{1}k$ etc.
Apply $C_3+10C_2+100C_1$
$\Delta =\left|\begin{array}{ccc}a& b& n_{1}k\\ l& m& n_{2}k\\ p& q& n_{3}k\end{array}\right|=k\parallel =k{\Delta }_1$
where ${\Delta }_1$ is a determinant consisting of integers.
$\therefore k{\Delta }_1$ is also an integer. Hence $\Delta$ is divisible by $k$