MGMAt Question Bank : Number Properties Q 18 of 25

[earthling]

Here is the problem from the MGMAT question bank

If x3 - x = n and x is a positive integer greater than 1, is n divisible by 8?

(1) When 3x is divided by 2, there is a remainder.

(2) x = 4y + 1, where y is an integer.

MGMAT states the correct answer is D. Explanation below:

However, for statement 1, if X-1, then the answer is No its not divisible by 8. For all other values of x, the answer is Yes. So why isnt the answer C? If anyone can point out any flaws in my logic i will appreciate it

Thanks

MGMAT Soln:

If we factor the equation in the question, we get n = x(x - 1)(x + 1) or n = (x - 1)x(x +1). n is the product of three consecutive integers. What would it take for n to be divisible by 8? To be divisible by 8, is to be divisible by 2 three times, or to have three 2's in the prime box.
The easiest way for this to happen is if x is odd. If x is odd, both x - 1 and x + 1 will be even or divisible by 2. Furthermore, if x is odd, x - 1 and x + 1 will also be consecutive even integers. Among consecutive even integers, every other even integer is divisible not only by 2 but also by 4. Thus, either x - 1 or x + 1 must be divisible by 4. With one number divisible by 2 and the other by 4, the product represented by n will be divisible by 8 if x is odd.
(1) SUFFICIENT: This tells us that x is odd. If 3x divided by 2 has a remainder, 3x is odd. If 3x is odd, x must be odd as well.
(2) SUFFICIENT: This statement tells us that x divided by 4 has a remainder of 1. This also tells us that x is odd because an even number would have an even remainder when divided by 4. Alternative method: if we rewrite this statement as x - 1 = 4y, we see that x - 1 is divisible by 4, which means that x + 1 is also even and the product n is divisible by 8.
The correct answer is D.

[jojo]

I am assuming you meant x=1 instead of x-1. I think you are right and the MGMAT answer is wrong. Could you any instructor reading this, pls clarify. Thanks

[idiot]

guys, the question says "If x3 - x = n and x is a positive integer greater than 1, is n divisible by 8?"
so x can't b equal to 1 & MGMAT answer/explanation is not wrong.[/code][/list]

[esledge]

If you did mean x=1, then "idiot" has your answer: the question stem eliminates that case.

However, from a mathematical point of view, x=1 isn't really an exception to the "is x odd?" rephrase of this question. If n = (x-1)(x)(x+1) and x = 1, then n = 0*1*2 = 0. Zero actually is divisible by 8, because 0/8 = 0, an integer. For that matter, zero is divisible by any non-zero integer. The GMAT doesn't tend to test this property, so that's probably why we specified "x is a positive integer greater than 1" here.

[DWG]

What level question is this on the GMAT?

[esledge]

This one is 600-700 in difficulty.

[Khalid]

If n = x^3-x
then n = (x-1) x (x+1)

With statement 1 we know x is odd and stem says x is >1

When x = 3 , n = 2x3x4 so answer is NO
When x = 5 , n = 4x5x6 so answer is No
when x = 7, n= 6x7X8- so answer is Yes

So isn't statement A insufficient?

[JonathanSchneider]

Khalid, you're making a mistake there. For a number to be divisible by 8, it needs to have three factors of 2. The above breakdown that you showed actually proves that we have three factors of 2 for each product. You don't need the 8 to come all at once. A 2 and a 4 will suffice.

Notice that whenever x is odd, you will have an even number on either side of it. Any pair of consecutive even numbers will include one number that is divisible by 4 and one that is not. Thus, we know that one of these evens will be divisible by 4, while the other will still be divisible by 2. That's our 8 right there (for the total product).

[malikrulzz]

If you did mean x=1, then "idiot" has your answer: the question stem eliminates that case.

However, from a mathematical point of view, x=1 isn't really an exception to the "is x odd?" rephrase of this question. If n = (x-1)(x)(x+1) and x = 1, then n = 0*1*2 = 0. Zero actually is divisible by 8, because 0/8 = 0, an integer. For that matter, zero is divisible by any non-zero integer. The GMAT doesn't tend to test this property, so that's probably why we specified "x is a positive integer greater than 1" here.

Ans should be B then

[JonathanSchneider]

No, D is in fact correct. Statement (1) is telling us that x is odd. This is sufficient.

[ColinT819]

I don't see how this is a 600-700 level question.

[RonPurewal]

I don't see how this is a 600-700 level question.

remember that 'difficulty level' is an aggregate statistic that actually has nothing to do with 'inherent difficulty' (whatever that even means, anyway).
if more people, and/or higher-scoring people, get an item incorrect, then it gets a higher 'level'. period. end of story.
most of the problems in the CAT software have been calibrated on the basis of thousands of students' results, so, while the results may be surprising, they're not groundless.

if it makes you feel any better, the data surprise me just as often as they probably surprise you.
in other words, even i can't reliably guess 'difficulty levels', except for the very extremes—and i've been teaching standardized tests for more than 20 years!

[RonPurewal]

in any case, there are some VERY important practical lessons that you should glean here.

takeaway:
you will NEVER be able to gauge 'difficulty ratings'... so don't bother.
really.
don't.
there will only be one meaningful outcome: you'll have less brainpower left for actually solving the problem. needless to say, this is not a desirable situation.

corollary:
DO NOT EVER think about 'difficulty levels'.
yep.
don't.
there's no point. you can't build a strategy on things you can't see.

[RonPurewal]

...at this point, you may be (rightly) wondering why our CAT exams are riddled with difficulty numbers. after all, ron just basically told you that they are pretty much useless for anyone who doesn't write tests.

if you are, in fact, wondering about this, there are at least 2 good reasons:

1/
they motivate some people:

this is an adaptive test.

the dastardly thing about adaptive tests is that, because they are adaptive, '% correct' does not generally improve as a test taker's performance improves.
for students who are fixated on numerical metrics, this can be disheartening.
(my advice to these students is, of course, not to be so concerned with numerical metrics—at least not until fairly late in the game—but we need to be realistic here.)

on the other hand, 'average difficulty rating' DOES improve as a test taker's performance improves... so, an increase in that number can be reassuring to these students.

motivation is VERY important.

2/
unfortunate business realities:
there's a HUGE market of students who want numbers numbers numbersnumbersnumbers (and even more numbers)... and a paucity of students who are relatively unconcerned with those numbers.
so, the fact is that the difficulty statistics increase our CAT exam revenues... and, unfortunately, we do have to sell stuff if we want to keep the lights on around here.

importantly, we do not capitulate to all, or even most, of the overall market's less desirable tendencies here. (e.g., if we published a 1000-page sentence correction book, we could probably charge a lot more for it, and we'd probably make more money from it. but, since we know that such a book would be MUCH less useful for our students... we don't.)
but the inclusion of difficulty levels is one area in which we decided that the benefits outweigh the risks. there you go, full disclosure.