If k and n are positive integers, is k a factor of 3n

You'll never need to worry about zero in divisibility questions on the GMAT anyway, and in this question (as in any divisibility question) the unknowns are restricted to positive integers. But in your example, you start off by positing that n is zero, but in your conclusion you seem to be instead assuming that it's k that is zero (we're asked if k is a factor of n, and you conclude by saying "since 0 is not a factor of any number", so I gather you're assuming at this point that it's k that is zero). So I think you've mixed up which number is zero in your example, and if you straighten that out, the answer will still make sense when n=0, though you won't need to worry about cases like that in any event.