Question

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

• A. divisible by $k$
• B. divisible by $k^2$
• C. divisible by $2k$
• D. None of these

Answer 1

Answer: (A)

Given, $A28$, $3B9$ and $62C$ are divisible by $k$
$\therefore A28=k$
$\Rightarrow 100A+20+8=k:3B9=k$
$\Rightarrow 300+10B+9=k$ and $62C=k$
$\Rightarrow 600+20+C=k$
Let $\Delta =\left|\begin{array}{ccc}A& 3& 6\\ 8& 9& C\\ 2& B& 2\end{array}\right|$
$=\frac{1}{100}\times \frac{1}{10}\left|\begin{array}{ccc}100A& 300& 600\\ 8& 9& C\\ 20& 10B& 20\end{array}\right|$
Applying $R_1\to R_1+R_2+R_3$
$=\frac{1}{1000}$
$\left|\begin{array}{ccc}100A+20+8& 300+10B+9& 600+20+C\\ 8& 9& C\\ 20& 10B& 20\end{array}\right|$
$=\frac{1}{1000}\left|\begin{array}{ccc}k& k& k\\ 8& 9& C\\ 20& 10B& 20\end{array}\right|$
$=\frac{k}{100}\left|\begin{array}{ccc}1& 1& 1\\ 8& 9& C\\ 20& 10B& 20\end{array}\right|$
Hence, it is always divisible by $k$