Great solutions.
This question can be done fairly quickly via backsolving and brute force (which is why I doubt that it's a real GMAT question).
On the GMAT, we'll always have 5 answer choices to help us out. First we use common sense to see if we can eliminate any choices. Even if we can't, we'd simply start with either B or D and have a 40% chance of getting it right on the 1st try and a 100% chance of getting it right on the 2nd.
For example, let's say the choices were:
A) 7, 17
B) 11, 19
C) 13, 23
D) 17, 29
E) 19, 31
Let's start with common sense: for D, the last 3 primes give us 19*23*29. Well, this is certainly more than 20^3, which is 8000, way too much (we want 7429). E is even bigger: eliminate D and E.
A, B and C are all in the realm of possibility (although you could also use common sense and a tiny bit of math to see that A is clearly going to be too small), so let's try B, the middle remaining choice:
11, 13, 17, 19
11*13*17 (a bit of brute force long multiplication never hurt anyone) = 2431. It's supposed to equal 4199, so clearly we need bigger numbers: eliminate A and B, choose C.
We also could have eliminated A and B just by focusing on the units digits:
For A, our first 3 numbers are 7, 11, 13. 7*1*3 ends in "1"; we want our product to end in "9".
For B, our first 3 numbers are 11, 13, 17. Again, 1*3*7 ends in "1"; we want our product to end in "9".
For C, our first 3 numbers are 13, 17, 19. 3*7*9 ends in "9"... bingo!
