Two numbers $x$ and  $y$  are chosen at random from the set  $\left\{1,2,3,\mathrm{.....30}\right\}.$ The probability that $x^2-y^2$  is divisible by 3 is

The total number of ways of choosing two numbers out of  $1,2,3,\mathrm{....30}$ is $30c_2=435$ Since $x^2-y^2$  is divisible by 3 if either none of a and b divisible by 3. or both are divisible by 3 Thus, the favorable number of cases $=10c_2+20c_2=235$

$\therefore$ Required probability $=\frac{235}{435}=\frac{47}{87}$