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 $\frac{3}{10}$.