Two non-negative integers are chosen at random from a set of non-negative integers  with replacement. If the probability that sum of the squares divisible by 10 is P then 50P is 

 $10\times 10=100 \mathrm{ways}.$
${\mathrm{x}}^2+{\mathrm{y}}^2$ is divisible by 10 if unit digit of sum of square of unit digit of x and y is zero
Unit digit of x, y can be (0,0), (1,3), (3,1), (1,7), (7,1), (2,4), (4,2), (2,6), (6,2), (3,9), (9,3), (4,8), (8,4), (5,5), (6,8), (8,6), (7,9), (9,7) 
$\therefore$18 cases are favourable, and 
$\begin{array}{l}\mathrm{P}=\frac{18}{100}=\frac{9}{50}\Rightarrow \\ \therefore 50\mathrm{P}=9\end{array}$