What is the remainder when (5^42 - 16^342*7^42) divided by 4 ?

Notice if you divide 19 by 4, the remainder is 3, but if you divide 29 by 4, the remainder is 1. So if we knew, say, that the units digit of some number was 9, we still wouldn't know what remainder we'd get when we divide the number by 4. So when you're dividing by 4 and need to find a remainder, it's not a units digit question. A number's units digit will only unambiguously tell you the remainder when you're dividing by 5 or by 10 (and technically by 2, but then you're just being asked if the number is even or odd).

The question in this thread isn't one that would ever appear on the GMAT, so there's no need to learn how to solve it. For one thing, to answer questions like this without doing quite a lot of work, you need to use math known as "modular arithmetic" (which could just be called "remainder arithmetic"). That field of math proves that we can, when dividing by 4, replace the base in "5^42" with any other number that has the same remainder as 5 has when we divide by 4. So we can replace the "5" with "9" or "41" or, most conveniently, "1", without changing the remainder. Similarly, we can replace the "16" in "16^342" with "0", and doing that, we just want to know the remainder when we divide 1^42 - 0^342 * 7^42 = 1 by 4, and the answer is 1. There are other ways to answer questions like this, though you either need to know some remainder arithmetic principles, or use the binomial theorem, which the GMAT will never expect any test taker to know.

And the question wouldn't appear on the test for another reason: it is also asking what remainder we get when we divide a negative number by 4. There isn't a consensus in math circles about how to calculate remainders when we divide negatives by positives, and if mathematicians don't even agree on what the answer to a question like this should be, the GMAT can't ask you what the answer should be. Most number theorists would say the answer here is 1 (and if you use modular arithmetic, as I did above, you're observing that definition of a remainder), but you'll see some definitions of remainders (especially if you read about math in computing science) that would say the answer here is -3. I'm not sure of the source of the question, but it's not a realistic GMAT problem.