If $D=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right|\text{, for }x\ne 0,y\ne 0$ then D is

Given, $D=\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+x& 1\\ 1& 1& 1+y\end{array}\right|$

Apply $R_2\to R_2-R_1\text{ and }R\to R_3-R_1$

$\therefore D=\left|\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}y\end{array}\right|=xy$

Hence, D is divisible by both x and y