Problem Solved. Thank you for your quick reply!
2. If neither a^2 nor b^2 is congrent to 0 modulo 3, then c^2 is congruent
to 2 modulo 3 which is not possible, hence at least one of a^2 or b^2 is
congrent to 0 modulo 3.
3. If x^2 is congruent to 0 modulo 3 then so is x, hence either a or b must
be congrent to 0 modulo 3.