Question

The digits A, B, C are such that the three digit numbers A88, 5B6, 86C are divisible by 72 then the determinant $\left|\begin{array}{ccc}A& 5& 8\\ 8& B& 6\\ 8& 6& C\end{array}\right|$ is divisible by

• A. $72$
• B. $144$
• C. $288$
• D. $216$

Answer 1

The correct option is A $72$

$A88=72x=100A+80+8$

$5B6=72y=500+10B+6$

$86C=72z=800+60+C$

Now let $\Delta =\left|\begin{array}{ccc}A& 5& 8\\ 8& B& 6\\ 8& 6& C\end{array}\right|=\frac{1}{100}\left|\begin{array}{ccc}100A& 500& 800\\ 8& B& 6\\ 8& 6& C\end{array}\right|$

Applying $R_1\to R_1+10R_2+R_3$

$\Delta =\frac{1}{100}\left|\begin{array}{ccc}100A+80+8& 500+10B+6& 800+60+C\\ 8& B& 6\\ 8& 6& C\end{array}\right|$

$=\frac{1}{100}\left|\begin{array}{ccc}72x& 72y& 72z\\ 8& B& 6\\ 8& 6& C\end{array}\right|$

$=\frac{72}{100}\left|\begin{array}{ccc}x& y& z\\ 8& B& 6\\ 8& 6& C\end{array}\right|$