Question

Let the three digit numbers $A28,3B9$, and $62C$, where $A,B,C$ are integers between $0$ and $9$, be divisible by a fixed integer $k$,. Show that the determinant $\left|\begin{array}{ccc}A& 3& 6\\ 8& 9& C\\ 2& B& 2\end{array}\right|$ is also divisible by the same integer $k$.

Answer 1

The numbers $A28,3B9,62C$ are divisible by $k$

Let $A28=ka,3B9=kb,62C=kC$ where $a,b,c$ are integers

Let $\Delta =\left|\begin{array}{ccc}A& 3& 6\\ 8& 9& C\\ 2& B& 2\end{array}\right|$

$=\left|\begin{array}{ccc}A& 3& 6\\ (8+100A+10.2)& (9+100.3+10.B)& (C+100.C+10.2)\\ 2& B& 2\end{array}\right|$

$\to$ [ By $(R_2+100R_1+10R_3)$]

$=\left|\begin{array}{ccc}A& 3& 6\\ A28& 3B9& 62C\\ 2& B& 2\end{array}\right|$

$=\left|\begin{array}{ccc}A& 3& 6\\ ka& kb& kc\\ 2& B& 2\end{array}\right|$

$=K\left|\begin{array}{ccc}A& 3& 6\\ a& b& c\\ 2& B& 2\end{array}\right|$

=integral multiple of $k$

$\therefore △$ is divisible by $k$