If z is an integer such that ||z - 30| - 43| = 62 which of the followi

I don't understand why there are pointless complications in the question. Remainders are never negative, so there's no reason to ask for "|r|", since that is always equal to r. Nor is there any reason to write 6^2 instead of 36 in the original absolute value equation. They'd only do that in a real GMAT question if they were trying to suggest to a test taker that there would be some advantage to thinking about that number as a square, instead of as 36, but there is no reason to think about 36 as if it were "6^2" in this problem.

Since |a - b| is the distance between a and b on the number line, the original absolute value equation tells us that |z-30| and 43 are separated by 36 on the number line. So either |z-30| is 36 greater than 43, and is equal to 79, or |z-30| is 36 less than 43, and is equal to 7. So we have two equations, and again using distances (or whatever absolute value method you prefer) :

|z - 30| = 79 --> so z is either 79 less than or 79 greater than 30, and z = -49 or 109
|z - 30| = 7 --> so z is either 7 less than or 7 greater than 30, and z = 23 or 37

We can now divide each solution by 7 and see what remainders are possible. You will absolutely never need to calculate remainders when dividing negative numbers by anything on the GMAT, so the first solution, z = -49, is one you'd never get in a GMAT question about remainders. But since that number is divisible by 7, it gives a remainder of 0. The number 109 is 105 + 4, and since 105 is divisible by 7, we get a remainder of 4. Since 23 and 37 both give a remainder of 2 when we divide by 7, the answer is I, II and III.