Two natural numbers x and y are chosen at random. What is the probability that `x^(2) + y^(2)` is divisible by 5?

Explanation

To solve the problem of finding the probability that \( x^2 + y^2 \) is divisible by 5 when two natural numbers \( x \) and \( y \) are chosen at random, we can follow these steps:

### Step 1: Understand the Remainders
When a natural number is divided by 5, the possible remainders (or residues) are 0, 1, 2, 3, and 4. We denote these remainders as \( r \) for \( x \) and \( s \) for \( y \).

### Step 2: Express \( x \) and \( y \)
We can express \( x \) and \( y \) in terms of their remainders:
- \( x = 5p + r \)
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