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Discrete Mathematics Level 4

Two numbers \(x,y\) with \(x>y\) are chosen at random from the set\(\left\{1,2,3,.....,299,300\right\}.\)

If the probability that \(x^{2} - y^{2}\) is divisible by 3 is \(\frac{a}{b},\) then what is the digit sum of \(a \times b\)?

Details and Assumptions:

  1. Digit sum refers to the sum of all the digits of the number. For example, the digit sum of 123 is \(1+2+3=6.\)
  2. \(a\) and \(b\) are co-prime numbers.

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