Data Sufficiency

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.

OA: D

Hi Richachampion,

This is a YES-NO DS question.

Question: Is n divisible by 8 ?

Where n = x^3 - x,

Remember a number to be divisible by 8, it should have minimum of three 2's.

Now lets little bit analyze "n" here,

Given, x^3-x=x*(x^2-1)=(x-1)*x*(x+1),

Notice that we have the product of three consecutive integers (It's given that x is a positive integer)

Now, if x=odd, then (x-1) and (x+1) are consecutive even integers, thus one of them will also be divisible by 4.

For example if x = 5 then 4 * 5 * 6 will have minimum three 2's

If x is odd then it is divisible by 8.

So we need check only whether x is odd.

Statement I is sufficient: When 3x is divided by 2, there is a remainder.

If x is even then 3x is divisible by 2, so the remainder is zero then which is not possible according to statement I.

So x has to be odd. So sufficient, where the remainder is one.

Statement II is sufficient: x = 4y + 1, where y is an integer.

Even + odd = Odd,

Here 4y is even and 1 is odd, so 4y+1 is always odd.

So x is odd.

So sufficient.

So the answer is D.

Here statement I is poorly worded (Don't know what's the source of the question) generally you don't see these type of official GMAC questions.

When 3x is divided by 2, there is a remainder - If we look at look it zero is also consider as a remainder, in that case x could be even as well. So it should be framed something like this "3x is not perfectly divisible by 2"

Hope this is clear.

crackverbal wrote: Here statement I is poorly worded (Don't know what's the source of the question) generally you don't see these type of official GMAC questions.

Source of the question is clearly mentioned in the Title. Thanks!

given x>1 and is an integer

=x­³ - x = n
=x(x-1)(x+1) = n => consecutive numbers
=(x-1)(x)(x+1)

let
x=2 = 1*2*3 = 6 (not divisible by 8)
x=3 = 2*3*4 = 24(divisible by 8)
x=4 = 3*4*5 = 60(not divisible by 8)
x=5 = 4*5*6 = 120(divisible by 8)
x=8 = 7*8*9 = 504(divisible by 8)

when x is odd or x=8,16(multiple of 8)
then x­³ - x will be divisible by 8

(1) When 3x is divided by 2, there is a remainder.
x is not divisible by 2 so its a odd number
SUFFICIENT

(2) x = 4y + 1, where y is an integer
x will be an odd number always (even + odd = odd)
SUFFICIENT

SO D