Hi MensaNumber,
This question can be solved rather easily by TESTing VALUES, although the work itself will take a bit longer than average and it would help a great deal if you could spot the subtle Number Properties involved.
From the question stem, you can see that we're dealing with division by 3 (or the 'rule of 3', if you learned the concept that way). You don't actually have to multiply out any of the answer choices though - you just need to find the one answer that will ALWAYS have a '3' in one of its 'pieces.' The subtle Number Property I referred to at the beginning is the 'spacing out' of the terms.
(1)(2)(3) is a multiple of 3, since it's 3 times some other integers.
(5)(6)(7) is also a multiple of 3, since we can find a 3 'inside' the 6, so we have 3x2 times some other integers.
Looking at the answer choices to this question, we're clearly NOT dealing with consecutive integers, but the 'cycle' of integers is something that we can still take advantage of.
For example, we know that...
When n is an integer, (n+1)(n+2)(n+3) will include a multiple of 3, since it's 3 consecutive integers (one of those 3 terms MUST be a multiple of 3, even if you don't know exactly which one it is).
You can take this same concept and 'move around' any (or all) of the pieces:
(n+1)(n+2)(n+6) will also include a multiple of 3 (that third term is 3 'more' than 'n+3').
Instead of adding a multiple of 3 to a term, you could also subtract a multiple of 3 from a term.
eg. (n-2)(n+2)(n+3) will also include a multiple of 3 (that first timer is 3 'less' than 'n+1').
The correct answer to this question subtracts a multiple of 3 from one of the terms.
Final Answer:
All things being equal, I'd still stick to TESTing VALUES (and not approaching the prompt with math theory) - the math is easy and you can put it 'on the pad' with very little effort.
GMAT assassins aren't born, they're made,
Rich