The integers x and y are chosen at random (with replacement) from all first digit naturalnumbers. Then find the probability that |x2 − y2| is divisible by 2.

First, let's understand the problem. We are choosing x and y randomly from the first digit natural numbers, which are 1 to 9. We need to find the probability that the absolute difference of their squares is divisible by 2.

Step 1: Identify the total number of outcomes Since we are choosing x and y from the set {1,2,3,4,5,6,7,8,9}, there are 9 choices for x and 9 choices for y. So, the total number of outcomes is 9*9 = 81.

Step 2: Identify the favorable outcomes We need to find the cases where |x^2 - y^2| is divisible by 2. This happens when x and y are both odd or both even. The reason is that the square of an odd number is odd, and the square of an even number is even. The difference between two odd numbers or two even numbers is always divisible by 2.

The odd numbers in the set {1,2,3,4,5,6,7,8,9} are {1,3,5,7,9}, so there are 5 choices for x and 5 choices for y when they are odd. So, there are 5*5 = 25 outcomes when x and y are odd.

The even numbers in the set {1,2,3,4,5,6,7,8,9} are {2,4,6,8}, so there are 4 choices for x and 4 choices for y when they are even. So, there are 4*4 = 16 outcomes when x and y are even.

So, the total number of favorable outcomes is 25 + 16 = 41.

Step 3: Calculate the probability The probability is the number of favorable outcomes divided by the total number of outcomes. So, the probability is 41/81.