This is a deceptively hard question. Grab some blank flash cards and get ready to learn. This is a classic theory problem, so if you happen to know the theory being tested here, that can be a really fast way to solve. But they do a great job of disguising what's being tested on this one, so if you don't recognize what it's testing, what do you do?
First, just make a mental note that if a value is divisible by 8, that value is also a multiple of 8. The values 8, 16, 24, 32, and so on, are all divisible by 8.
Since this is a theory problem, try real numbers to see whether you can figure out the theory. The problem specifies that x is a positive integer greater than 1, so x = 2 or greater. For example:
If x = 2, then x3 – x = 8 – 2 = 6 = n. In this case, the answer to the question is No, n is not divisible by 8.
If x = 3, then x3 – x = 27 – 3 = 24 = n. In this case, the answer to the question is Yes, n is divisible by 8.
It would be annoying to keep going much further because the numbers will get really big. Look at the given equation again to see whether it can be rearranged in any way:
x3 – x = n
x(x2 – 1) = n
x(x + 1)(x – 1) = n
When you take an integer and add 1 to it to get another integer, you have two consecutive integers. In fact, the expression in this equation represents three consecutive integers: (x – 1), x, and (x + 1). And those three consecutive integers multiply together to give you n. (Now that you know this, grab a flash card and record the fact that the expression x3 – x is really telling you that you have three consecutive integers. Next time, you won't have to figure it out; you'll just recognize it.)
Write out some consecutive integers and examine:
2, 3, 4, 5, 6, 7, 8, 9, 10
If the consecutive set is (1)(2)(3), it's not divisible by 8.
If the consecutive set is (2)(3)(4), it's divisible by 8, because it contains (2)(4).
If the consecutive set is (3)(4)(5), it's not divisible by 8.
If the consecutive set is (4)(5)(6), it's divisible by 8, because it contains (4)(6), which can be broken down into (4)(2)(3).
Don't take the time to figure all of this out during the GMAT; learn it beforehand (right now!) and then apply your knowledge when it pops up again on the test. Here's the full pattern:
First, if the consecutive set is odd-even-odd, then the middle value itself has to be a multiple of 8 in order for the product to be a multiple of 8—for example, (7)(8)(9). But if that middle value is not a multiple of 8 (as was the case in the two odd-even-odd examples shown earlier), then this whole product will not be a multiple of 8.
Second, if the consecutive set is even-odd-even, then the first and third values combined can create a multiple of 8. And, it turns out, any two consecutive even values will always multiply to a multiple of 8, because one of the integers will always be a multiple of 2 and the other will always be a multiple of 4. Check it out:
2, 4, 6, 8, 10, 12, ...
(2)(4) = multiple of 8
(4)(6) = multiple of 8
(6)(8) = multiple of 8
This goes on forever, to infinity. So, if you know that the consecutive set is even-odd-even, then you know that the product is always a multiple of 8. Put that on a flash card too! Now, let's solve this problem.
(1) SUFFICIENT: This statement specifies that 3x divided by 2 has a remainder that isn't zero. If you divide an even number by 2, you'll always have a remainder of 0, so 3x must be an odd number. Further, if you multiply any integer by an even integer, you'll always get an even number, so x itself must be an odd number in order for 3x to be odd. (Try a couple of real numbers to prove that to yourself.)
Since x must be odd, the three consecutive integers in this case are even-odd-even. For the reasons discussed earlier, an even-odd-even consecutive pattern will always be a multiple of 8 and will therefore always be divisible by 8. This statement is sufficient to answer the question: Yes, n is always divisible by 8.
(2) SUFFICIENT: This statement introduces a new variable! If x = 4y + 1, where y is an integer, what does that signify? If you multiply any integer by 4, you'll end up with an even integer (because any integer multiplied by an even integer always returns an even integer). If you take any even integer and add 1 to it, you'll end up with an odd integer. So this statement is providing the same information as statement (1): The variable x must be odd.
As before, since x must be odd, the three consecutive integers in this case are even-odd-even, and an even-odd-even consecutive pattern will always be a multiple of 8 and be divisible by 8. This statement is sufficient to answer the question: Yes, n is always divisible by 8.
The correct answer is (D): Either statement, by itself, is sufficient to answer the question.
The big takeaway: During your studies, learn these "codes" that the GMAT likes to use to disguise information:
x3 – x really means that you have three consecutive integers multiplied together.
The product of any three consecutive integers in the pattern even-odd-even will always be divisible by 8 (and by 2 and by 4). (And it turns out that the product of any three consecutive integers will also always be divisible by 3. Play around with some numbers to try to figure out why!)
