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Is (x)(x + 2)(x + 4) divisible by 12?

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Is (x)(x + 2)(x + 4) divisible by 12?

by gmattesttaker2 » Wed May 14, 2014 9:12 pm
Hello,

Can you please tell me where I am going wrong here:

If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12?

1) x^2 + 2x is a multiple of 3.
2) 3x is a multiple of 2.

OA: B

For (x)(x + 2)(x + 4) to be divisible by 12, x should have at-least two 2's and one 3.

1) x^2 + 2x is a multiple of 3.
=> x(x + 2) is a multiple of 3
=> x has at-least one 3

In-suff.


2) 3x is a multiple of 2
=> x has at-least one 2

Hence, in-sufficient?


Can you please tell me where I am going wrong? Thanks for your help.


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Sri
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by Tushar14 » Wed May 14, 2014 9:28 pm
Hello Sri,
Whether x is even or odd, out of x, x+2 and x+4 one must be a multiple of 3. So, we need to find whether (x)(x + 2)(x + 4) is divisible by 4. Now, if x=odd, then all three multiples are odd, thus (x)(x + 2)(x + 4) will be odd and not divisible by 4. If x=even, then (x)(x + 2)(x + 4)=even*even*even, thus it'll be divisible by 4.

So to get to conclusion we need to track whether or not X is even;

Statement 1:- If we put X=1 (odd) then answer is NO, but if X=4 (even) then YES....Insufficient

Statement 2:- 3x is a multiple of 2. It clearly implies that x=even. Sufficient.

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by theCodeToGMAT » Thu May 15, 2014 8:08 am
To find: (x) (x+2) (x+4)/12

Statement 1:
x(x+2) = 3 * _ ==> this doesn't tell us whether we will have "4" in the numerator
INSUFFICIENT

Statement 2:
3x = 2 * _ ==> the minimum value 6 ==> x = 2
So, 2 * 4 * 6 / 12 = Yes
The next possible value will be 12.. so this pattern will carry on
SUFFICIENT
[spoiler]
{B}[/spoiler]
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by ceilidh.erickson » Thu May 15, 2014 9:18 am
Whenever you see a structure like (x)(x + 2)(x + 4), it's really testing products of CONSECUTIVE INTEGERS. Some other similar formats:

(n - 1)(n)(n + 1)
x(x - 1)(x - k)
etc

When we're asked about divisibility with products of consecutives, think about what must always be true, and what's variable in the particular case.

(x)(x + 2)(x + 4) is either three evens in a row, or three odds in a row. In either case, we're definitely going to have one multiple of 3. Out of any 3 consecutive integers, one of them will be a multiple of 3. The same is true for 3 consecutive evens, or 3 consecutive odds.

If we want to know if the product is divisible by 12, and we already know that it must be divisible by 3, the only remaining question is - is it divisible by two 2's? If all of those integers are even, it certainly will be. If they're odd, it won't be.

Rephrased question: Is x even?

1) x^2 + 2x is a multiple of 3.
Factor this:
x(x + 2) is a multiple of 3.
Ok, so one of those two terms is the multiple of 3, but that doesn't help us to answer the question of even or odd.
Insufficient.

2) 3x is a multiple of 2.
If 3x is a multiple of 2, then x must be a multiple of two (thus even). If x is even, so is x + 2 and x + 4, meaning that we have a product of 3 evens.
Sufficient.
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by ceilidh.erickson » Thu May 15, 2014 9:21 am
gmattesttaker2 wrote:Hello,

Can you please tell me where I am going wrong here:

If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12?

1) x^2 + 2x is a multiple of 3.
2) 3x is a multiple of 2.

OA: B

For (x)(x + 2)(x + 4) to be divisible by 12, x should have at-least two 2's and one 3.

1) x^2 + 2x is a multiple of 3.
=> x(x + 2) is a multiple of 3
=> x has at-least one 3
To answer more specifically your question about where you're going wrong... you're diving into the statements too quickly! You parsed "divisible by 12" correctly, but you didn't dissect/analyze the (x)(x + 2)(x + 4) structure before you dove straight into statement 1.

The way to master DS questions is always to unpack the question stem as completely as possible before addressing the statements.
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by ceilidh.erickson » Thu May 15, 2014 9:23 am
For more on consecutive products, see:
https://www.beatthegmat.com/totaly-lost- ... tml#716315
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by GMATGuruNY » Thu May 15, 2014 10:23 am
gmattesttaker2 wrote: If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12?

1) x^2 + 2x is a multiple of 3.
2) 3x is a multiple of 2.
Statement 1: x² + 2x is a multiple of 3
Case 1: x=1, implying that x² + 2x = 3
In this case, (x)(x + 2)(x + 4) = 1*3*5, which is not a multiple of 12.
Case 2: x=12, implying that x² + 2x = 168
In this case, (x)(x + 2)(x + 4) = 12*14*16, which is a multiple of 12.
INSUFFICIENT.

Statement 2: 3x is a multiple of 2.
Options for 3x:
2, 4, 6, 8, 10, 12, 14, 16, 18...
Dividing by 3, we get the following options for x:
2/3, 4/3, 2, 8/3, 10/3, 4, 14/3, 16/3, 6...

Since x must be an INTEGER, only the following options are viable:
2, 4, 6, 8, 10, 12...
If x=2, then (x)(x + 2)(x + 4) = 2*4*6, which is a multiple of 12.
If x=4, then (x)(x + 2)(x + 4) = 4*6*8, which is a multiple of 12.
If x=6, then (x)(x + 2)(x + 4) = 6*8*10, which is a multiple of 12.

In every case, (x)(x + 2)(x + 4) = multiple of 12.
SUFFICIENT.

The correct answer is B.
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