Video Explanation

Text Explanation

Plugging in numbers will result in numbers way too big.  Instead, let’s factor that algebraic expression.

Q = x³ − x = x(x² − 1) = x(x + 1)(x − 1)

Notice that we factored out a factor of x, then used the Difference of Two Squares formula, which is discussed in a GRE blog and in a GMAT blog.  Rewrite the order of the factors.  

Q = (x − 1)x(x + 1)  

If x is a positive integer, then (x − 1) is the integer immediately before it, and (x + 1) is the integer immediately after it.  Thus, Q is the product of three consecutive integers.

We know that, in any group of three consecutive integers, either one is even and two are odd, or one is odd and two are even.  Thus, it’s assured that at least one is even, and so Q is definitely divisible by 2. We also know that, in any group of three consecutive integers, there must be exactly one factor of three, so Q is definitely divisible by 3.  Any number that is simultaneously divisible by 2 and by 3 has to be divisible by 6, so Q must be divisible by 6.

Since 6 goes evenly into Q, the remainder is zero.  

Answer = (A)