Probability that $5|(x^2+y^2)$,where $x,y\in\Bbb{N}$ Question:
Two numbers $x,y$ are selected from the set of natural numbers. Then what is the 
probability that $x^2+y^2$ is divisible by $5$?
Attempt:
Since the natural number upper limits are not given, I do not understand how can I divide the number in parts.
mean $1,6,11,16,\cdots \cdots 5n+1$
our $2,7,12,17,\cdots \cdots 5n+2$
our $3,8,13,18,\cdots \cdots 5n+3$
our $4,9,14,19,\cdots \cdots 5n+4$
our   $5,10,15,20,25,\cdots \cdots, 5n$ 
Could some one help me with this? Thanks.

Take the set of all possible unit digits of the squares of natural numbers as $\{0, 1, 4, 5, 6, 9\}$. But however all aren't equally likely: $0$ and $5$ only occur with probability $\frac{1}{10}$ each, the others with $\frac{1}{5}$ each. Now weighting all of the possible squares modulo $10$ by these probabilities, we get $\frac{9}{25}$ as the probability. Hope it helps. 
