Brian@VeritasPrep wrote:
So for this problem, statement 1 tells us that x^3 + x is divisible by 4. Well, that means that it's even.
But that's not *all* it means. If you rephrase the statement "x^3 + x is divisible by 4" as "x^3 + x is even", you're losing a lot of information, so much information that you'll get the wrong answer here.
Brian@VeritasPrep wrote:
I think the biggest key here is to recognize up front that the question is asking "is x EVEN?". From there, "...divisible by 4"; "...divisible by 6" and statements like that are really only worthwhile to you as indications of even or odd, so you don't have to do the algebra...you can just test the number properties.
No, this is not the case. You can see this easily enough by taking a very simple example. If I ask if x is even, and tell you that x + x is divisible by 4, then 2x must be divisible by 4, so x must be divisible by 2. You would get the wrong answer if you rephrased the statement 'x + x is divisible by 4' as 'x + x is even', since then you would think x might be odd.
The same thing happens in the question above. We know from Statement 1 that x(x^2 + 1) is not only even, but is divisible by 4. Since one of x or x^2+1 is odd, the other even, we know that either x is divisible by 4, or that x^2 + 1 is divisible by 4. It's certainly possible for x to be divisible by 4, in which case x is even. It is absolutely impossible, however, for x^2 + 1 to be divisible by 4. For that to be true, x would need to be an odd number. If x is an odd number, then x^2 - 1 = (x+1)(x-1) is always equal to the product of two even numbers, so is always divisible by 4 (in fact, it is always divisible by 8, since it's the product of two consecutive even numbers, and among two consecutive even numbers you always have one multiple of 4). Since when x is odd, x^2 - 1 and x^2 + 1 are consecutive even numbers, only one of them can be divisible by 4. Since x^2 - 1 is divisible by 4, x^2 + 1 is never divisible by 4. So from Statement 1, it's impossible for x to be odd, and Statement 1 is sufficient.
