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).
[Math] Probability of selecting two numbers with a sum of squares divisible by 10
probability
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Best Answer
Your question seems to be more about the correctness of the problem in the first place.
The problem is incorrectly posed. There is no sample space and probability distribution that would make this problem "well posed".
Rather, without warning, what is happening here is different. These problems, without you being told, mean the following:
FIRST, take a very large $N$ and solve the problem for natural numbers picked from $0$ to $N$. Compute the probability, call it $p_N$. Then compute the limit of $p_N$ as $N \to \infty$.
You may be able to solve the problem for $N$ of the form $10k + 9$ (so the number of natural numbers from $0$ to $N-1$ is a multiple of $10$). In that case the answer may actually not depend on $N$. If you don't fuss about all the details, you may claim that that's the answer.