Official Explanation:
For this one, we’ll at least start out with picking numbers and see how far we get with this strategy.
Statement #1: Q = \(K^2\)
For simplicity, say that K = 20, so Q = 400. If Z = 40, then it goes evenly into 400, so the remainder is zero. On the other hand, if Z = 250, then it goes once into 400 with a remainder of 150. Two different choices of numbers give two different values of the remainder. This choice, alone and by itself, is not sufficient.
Statement #2: Z = K + 1
For simplicity, let K = 10. This makes Z = 11, but that doesn’t matter. Under this statement, there’s no restriction on the value of Q. If Q = 60, then 10 divides evenly into 60, and the remainder is zero. If Q = 69, then 10 goes into it six times with a remainder of 9. Two different choices of numbers give two different values of the remainder. This choice, alone and by itself, is not sufficient.
Combined statements:Let’s pick K = 11. Then Z = 12 and Q = 121. We know that 12 goes evenly into 120, so when we divide 121 by 12, we get a remainder of 1.
It gets hard to pick numbers and do the squaring & division without a calculator. All other choices will result in the same remainder of 1. Remember that we can use picking numbers to disqualify an answer, to show that it leads to two different conclusions, but if the same numerical answer results every time, we need to verify that with algebra or logic.
Remember the Difference of Two Squares pattern. We know that
Q − 1 = \(K^2\)− 1 = (K + 1)(K − 1)
Thus, Z = K + 1 always divides evenly into Q − 1. If we add one, the next integer up from Q − 1 is Q—the integer Q is exactly one more than the integer Q − 1. When we divide Q by Z = K + 1, Z always goes evenly into Q − 1, and the extra 1 is left over, so we always will have a remainder of 1.
The two statements together allow us to find a unique numerical answer to the prompt questions. Combined, the statements are sufficient.
Answer = (C)Thanks So much.
I got it totally wrong.