If n and t are positive integers, what is the greatest prime factor of nt?

1). The greatest common factor of n and t is 5

2). The least common multiple of n and t is 105

GMATPREP question. What is the best approach to this problem?

consider the 'prime box' approach (= an imaginary box that contains all the numbers in the prime factorization of a number, for those of you who are uninitiated into our curriculum).

you're looking for the greatest prime # that would be in the 'box' obtained from dumping all the factors of n and all the factors of t, including all repetitions, into a bigger box. (this is what multiplication does: it multiplies the complete factorization of one number by that of another. for instance, 12 = 2x2x3 and 20 = 2x2x5, so 12 x 20 = 2x2x2x2x3x5.) therefore, the question can be rephrased as follows: what is the greatest prime # that is a factor of either t or n ?

(1) this only tells is that the greatest number that is in both factorizations - those of n and t - is 5. but there could be a larger factor that is part of only one of the factorizations. for instance:
- it's possible that n = t = 5. then the greatest prime factor of nt is 5.
- it's possible that n = 5 and t = 35. then the greatest prime factor of nt is 7.
insufficient.

(2) the least common multiple contains every factor of t or n at least once. (it has to; if, say, t had a factor that wasn't contained in it, then it would fail to be a multiple of t.) so, the biggest prime factor of this # will also be the biggest prime factor of the product nt.
sufficient.
try a few combinations of n and t if you aren't convinced.

answer = b

In this case, is the greatest prime factor of the product of nt equal to 7?

yeah. it's whatever is the greatest prime factor of 105, as explained above.
because the prime factorization of 105 is 3*5*7, the answer is 7.

remember that this is a data sufficiency problem, though -- i.e. there is actually no need to generate the answer explicitly, as long as you have established that you have enough information to find it. (although, of course, doing so is good practice in case you see a similar problem in problem solving.)