Ok. WHen n is divided by m, we get an answer and a remainder of 3.
Thus, the answer here is p, which is a randomly assigned variable, with a remainder of 3.
This is where they get n = mp+3
(n is equal to m times p with 3 more added.)
q is another randomly assigned variable, and since we know that m is a multiple of 4, we know that m is 4 times something, or 4q.
Then, we can use our original equation: n = mp + 3 and replace the m with 4q, leaving us with:
n = 4pq + 3
Then they start picking values for p and q, beginning with 1 for both, and then 1 for one of them and 2 for the other to find our smallest values...
This *could* be an English translation issue, or more likely, it's a product of the fact that a lot of this math is rusty for those of us starting to study for the GMAT, and the approach we use may be slightly different than the approach to which you're accustomed. Start into the class, work on the processes they give you, and see if it gets better. Practice is your best friend here as far as making sense of this test. The pre-test is in no way a prediction of how well you'll do on the test. Rather, it's a gauge for you to see where you might need to work, and a tool to get you used to doing math problems in a test-like setting again.