Question

If two numbers \(x\) and \(y\) are chosen one after the other at random with replacement from the set of numbers \( \{1, 2, 3, \ldots, 10\} \), then the probability that \( |x^2 - y^2| \) is divisible by 6 is:

• A. \( \frac{8}{25} \)
• B. \( \frac{6}{25} \)
• C. \( \frac{3}{10} \)
• D. \( \frac{13}{50} \)

Hint:

When working with divisibility problems, try to break down the expression into factors that can be divisible by smaller numbers.

Answer 1

The Correct Option is C

We need to find the probability that \( |x^2 - y^2| \) is divisible by 6, where \( x \) and \( y \) are chosen from \( \{1, 2, 3, \ldots, 10\} \). We use the factorization \( x^2 - y^2 = (x - y)(x + y) \), and for this expression to be divisible by 6, either \( x - y \) or \( x + y \) must be divisible by 2 and 3.
Through calculation, you can verify that the total number of favorable outcomes is 30, and the total number of possible outcomes is 100. 
Therefore, the probability is: \[ \frac{30}{100} = \frac{3}{10}. \] Thus, the correct answer is \( \frac{3}{10} \).