<s>[INST] Probability of selecting two numbers with a sum of squares divisible by 10 
Two natural numbers $x$ and $y$ are chosen at random.  Find the probability that $x^2 + y^2$ is divisible by 10.

I could not understand how select two numbers from any natural number (infinite). 
 [/INST] Its simple logic. Count numbers on the basis of last digits whose sum of square makes 0.
So Favourable cases = {(0,0),(1,3),(1,7),(2,4),(2,6),(3,1),(3,9),(4,2),(4,8),(5,5),(6,2),(6,8),(7,1),(7,9),(8,4)(8,6),(9,3),(9,7)} = 18
Total cases = Value of x can be any digit from 0 to 9. 
Similarly for y.
$= 10 * 10 = 100$
$Probability = \frac{\text{No. of favourable cases}}{\text{Total no. of cases}}$
$= \frac{18}{100} = \frac{9}{50} $
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