25. Alvita is planning a garden patio to be made from identical square paving stones laid out in a rectangle measuring $x$ stones by $y$ stones. She finds that when she adds a border of width one stone around the patio, the area of the border is equal to the original area of the patio. How many possible values for $x$ are there?
A 1
B 2
C 4
D 8
E 16
25. C Let the stones measure 1 unit by 1 unit.
Then, after the border is added, the patio measures $x+2$ by $y+2$.
So the area of the border is $(x+2)(y+2)-x y$.
Therefore $(x+2)(y+2)-x y=x y$. So $x y+2 x+2 y+4-x y=x y$, that is $x y-2 x-2 y=4$. Hence $x y-2 x-2 y+4=8$, that is $(x-2)(y-2)=8$.
The table shows possible integer values of $x-2$ and the corresponding values of $y-2, x$ and $y$.
\begin{tabular}{ccrr}
$x-2$ & $y-2$ & $x$ & $y$ \\
\hline-8 & -1 & -6 & 1 \\
-4 & -2 & -2 & 0 \\
-2 & -4 & 0 & -2 \\
-1 & -8 & 1 & -6 \\
1 & 8 & 3 & 10 \\
2 & 4 & 4 & 6 \\
4 & 2 & 6 & 4 \\
8 & 1 & 10 & 3
\end{tabular}

Note that both $x$ and $y$ must be positive, so there are four possible values of $x$, namely $3,4,6,10$.