Two numbers a and b are chosen at random from the set of first 30 natural numbers. The probability that a2 – b2 is divisible by 3 is:

1) 9/87
2) 15/87
3) 47/87
4) 34/87

• Note that a2−b2 is divisible by 3 if and only if either (i) a and b are both divisible by 3 or (ii) neither a nor b is divisible by 3. This is because if n is not divisible by 3, then n has remainder 1 or 2 on division by 3. If a and b have the same remainder on division by 3, then 3 divides a−b. And if one has remainder 1 and the other 2, then 3 divides a+b. Finally, a2−b2=(a−b)(a+b).
  
  There are (10C2) choices of Type (i), and (20C2) of Type (ii).
  
  Alternately, we count the bad pairs, where one of the numbers is divisible by 3 and the other is not. There are (10)(20) bad pairs.
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