174682 works here, though I don't see any especially fast way to find it. It's helpful to notice that we have nine predictions, and if one digit is correct in each, then the answer must have three digits which appear in two predictions, and three digits which appear in one prediction. I then noticed that the first and last digits are paired in each prediction except when the last digit is 2 (by paired, I mean that when the first digit is 1, the last digit is 6, and when the first digit is 6, the last digit is 1), and anticipating that this restriction would be what would prevent most possibilities from working, I then settled on 2 for the last digit, and found the remaining digits through a process of elimination.

That certainly took more than two minutes, and while there may well be a faster way to get to the answer directly, I don't see it at a glance. If it were a real GMAT problem (it is not at all similar to any GMAT problem I've ever seen), and if it had answer choices, then, as Stuart points out, it would be straightforward to answer.