It is currently 19 Nov 2017, 11:23

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# If x is an integer, then x(x – 1)(x – k) must be evenly divi

Author Message
TAGS:

### Hide Tags

Manager
Joined: 07 Feb 2010
Posts: 155

Kudos [?]: 768 [1], given: 101

If x is an integer, then x(x – 1)(x – k) must be evenly divi [#permalink]

### Show Tags

15 Dec 2010, 07:19
1
KUDOS
15
This post was
BOOKMARKED
00:00

Difficulty:

45% (medium)

Question Stats:

64% (01:20) correct 36% (02:08) wrong based on 619 sessions

### HideShow timer Statistics

If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT

A. -4
B. -2
C. -1
D. 2
E. 5
[Reveal] Spoiler: OA

Last edited by Bunuel on 09 Jul 2013, 09:59, edited 1 time in total.
Renamed the topic and edited the question.

Kudos [?]: 768 [1], given: 101

Senior Manager
Status: Bring the Rain
Joined: 17 Aug 2010
Posts: 390

Kudos [?]: 47 [0], given: 46

Location: United States (MD)
Concentration: Strategy, Marketing
Schools: Michigan (Ross) - Class of 2014
GMAT 1: 730 Q49 V39
GPA: 3.13
WE: Corporate Finance (Aerospace and Defense)
Re: x(x – 1)(x – k) [#permalink]

### Show Tags

15 Dec 2010, 07:41
There is probably an easier way, but I just used the picking numbers option for this.

I chose x=2
2(1)(2-k) then just plugged in the answer choices for K until one wasn't evenly divisible by 3.

B gives you 8. 8/3 is not an integer.

_________________

Kudos [?]: 47 [0], given: 46

Math Expert
Joined: 02 Sep 2009
Posts: 42257

Kudos [?]: 132703 [10], given: 12335

Re: x(x – 1)(x – k) [#permalink]

### Show Tags

15 Dec 2010, 07:52
10
KUDOS
Expert's post
6
This post was
BOOKMARKED
anilnandyala wrote:
If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT

-4
-2
-1
2
5

We have the product of 3 integers: (x-1)x(x-k).

Note that the product of 3 integers is divisible by 3 if at least one multiple is divisible by 3. Now, to guarantee that at least one integer out of x, (x – 1), and (x – k) is divisible by 3 these numbers must have different remainders upon division by 3, meaning that one of them should have remainder of 1, another reminder of 2 and the last one remainder of 0, so be divisible by 3.

Next, if k=-2 then we'll have (x-1)x(x+2)=(x-1)x(x-1+3) --> which means that (x-1) and (x+2) will have the same remainder upon division by 3. Thus for k=-2 we won't be sure whether (x-1)x(x-k) is divisible by 3.

30 second approach: 4 out of 5 values of k from answer choices must guarantee divisibility of some expression by 3. Now, these 4 values of k in answer choices must have some pattern: if we get rid of -2 then -4, -1, 2, and 5 creating arithmetic progression with common difference of 3, so -2 is clearly doesn't belong to this pattern.

Hope it helps.
_________________

Kudos [?]: 132703 [10], given: 12335

Manager
Joined: 10 Nov 2010
Posts: 157

Kudos [?]: 336 [0], given: 6

Re: x(x – 1)(x – k) [#permalink]

### Show Tags

03 Jan 2011, 22:00
Hi bunuel,
I don't understand the problem language, it says

If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT

how does it matter whats the value of K, i can choose x = 3 and the expression will always be divisible by 3.

Am i missing any minor yet important point?

Kudos [?]: 336 [0], given: 6

Math Expert
Joined: 02 Sep 2009
Posts: 42257

Kudos [?]: 132703 [0], given: 12335

If x is an integer, then x(x – 1)(x – k) must be evenly divi [#permalink]

### Show Tags

04 Jan 2011, 03:22
vjsharma25 wrote:
Hi bunuel,
I don't understand the problem language, it says

If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT

how does it matter whats the value of K, i can choose x = 3 and the expression will always be divisible by 3.

Am i missing any minor yet important point?

Stem says: "If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT"

The important word in the stem is "MUST", which means that we should guarantee the divisibility by 3 no matter the value of x (for ANY integer value of x), so you cannot arbitrary pick its value.

Hope it's clear.
_________________

Kudos [?]: 132703 [0], given: 12335

Manager
Joined: 10 Nov 2010
Posts: 157

Kudos [?]: 336 [0], given: 6

Re: x(x – 1)(x – k) [#permalink]

### Show Tags

04 Jan 2011, 08:46
OK. Now i get it.

Thanks Bunuel.

Kudos [?]: 336 [0], given: 6

Intern
Joined: 11 Nov 2010
Posts: 5

Kudos [?]: 8 [0], given: 1

Re: x(x – 1)(x – k) [#permalink]

### Show Tags

04 Jan 2011, 09:41
To be divisible by 3, one of these sequences must be divisible by 3.

X(X-1) (X-k)

Any 3 sequence number will always be divisible by 3. So X(X-1) (x-2) is divisible by 3.

K = 2, divisible by 3
K= 5, also a sequence ( parallel ) divisible by 3
K= -1, sequence is (X-1) X (X+1) so divisible by 3
K= -4, also a sequence ( parallel ) divisible by 3
K=-2, not a sequence, may not be divisible by 3

Kudos [?]: 8 [0], given: 1

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7736

Kudos [?]: 17797 [5], given: 235

Location: Pune, India
Re: x(x – 1)(x – k) [#permalink]

### Show Tags

04 Jan 2011, 19:49
5
KUDOS
Expert's post
7
This post was
BOOKMARKED
anilnandyala wrote:
If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT

-4
-2
-1
2
5

I am providing the theoretical explanation below. Once you get it, you can solve such questions in a few seconds in future!

Notice a few things about integers:
-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16......

Every number is a multiple of 1
Every second number is a multiple of 2
Every third number is a multiple of 3
Every fourth number is a multiple of 4 and so on...

So if I pick any 3 consecutive integers, one and only one of them will be a multiple of 3: e.g. I pick 4, 5, 6 (6 is a multiple of 3) or I pick 11, 12, 13 (12 is a multiple of 3) etc..

x(x - 1)(x - k) will be evenly divisible by 3 if at least one of x, x - 1 and x - k is a multiple of 3. We know from above, (x - 2)(x - 1)x will have a multiple of 3 in it. Also, (x-1)x(x + 1) will have a multiple of 3 in it because they both are products of 3 consecutive integers. So k can be 2 or -1. Eliminate these options.
Now let me write down consecutive integers around x:

(x-5), (x - 4), (x - 3), (x - 2), (x - 1), x, (x + 1), (x + 2), (x + 3), (x + 4), (x + 5) etc

(x - 2)(x - 1)x will have a multiple of 3 in it. x could be the multiple of 3, (x - 1) could be the multiple of 3 or (x - 2) could be the multiple of 3, in which case (x - 5) will also be a multiple of 3.
So in any case, (x - 5)(x - 1)x will have a multiple of 3 in it. So k can be 5.

Similarly, (x-1)x(x + 1) will have a multiple of 3 in it. x could be the multiple of 3, (x - 1) could be the multiple of 3 or (x + 1) could be the multiple of 3, in which case (x + 4) will also be a multiple of 3.
So in any case, (x - 1)x(x + 4) will have a multiple of 3 in it. So k can be -4.

We cannot say whether (x-1)x(x + 2) will have a multiple of 3 in it and hence if k = -2, we cannot say whether the product is evenly divisible by 3.

_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Kudos [?]: 17797 [5], given: 235

Senior Manager
Affiliations: SPG
Joined: 15 Nov 2006
Posts: 320

Kudos [?]: 894 [0], given: 28

Re: x(x – 1)(x – k) [#permalink]

### Show Tags

05 Jan 2011, 01:29
a. -4
b. -2 [2 more than A]
c. -1 [3 more than A]
d. 2 [6 more than A]
e. 5 [9 more than A]

nice, so we do have a pattern ... 4 answers have a difference of a multiple of 3 except B ... 3, 6, 9 are all multiples of 3

so we can select B without solving much

_________________

press kudos, if you like the explanation, appreciate the effort or encourage people to respond.

Kudos [?]: 894 [0], given: 28

Manager
Joined: 02 Oct 2010
Posts: 145

Kudos [?]: 50 [0], given: 29

Re: x(x – 1)(x – k) [#permalink]

### Show Tags

07 Jan 2011, 23:49
Bunuel wrote:
anilnandyala wrote:
If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT

-4
-2
-1
2
5

We have the product of 3 integers: (x-1)x(x-k).

Note that the product of 3 integers is divisible by 3 if at least one multiple is divisible by 3. Now, to guarantee that at least one integer out of x, (x – 1), and (x – k) is divisible by 3 these numbers must have different remainders upon division by 3, meaning that one of them should have remainder of 1, another reminder of 2 and the last one remainder of 0, so be divisible by 3.

Next, if k=-2 then we'll have (x-1)x(x+2)=(x-1)x(x-1+3) --> which means that (x-1) and (x+2) will have the same remainder upon division by 3. Thus for k=-2 we won't be sure whether (x-1)x(x-k) is divisible by 3.

30 second approach: 4 out of 5 values of k from answer choices must guarantee divisibility of some expression by 3. Now, these 4 values of k in answer choices must have some pattern: if we get rid of -2 then -4, -1, 2, and 5 creating arithmetic progression with common difference of 3, so -2 is clearly doesn't belong to this pattern.

Hope it helps.

Bunnel,
The second approach is too good...

Kudos [?]: 50 [0], given: 29

Non-Human User
Joined: 09 Sep 2013
Posts: 15656

Kudos [?]: 282 [0], given: 0

Re: If x is an integer, then x(x – 1)(x – k) must be evenly divi [#permalink]

### Show Tags

29 Nov 2013, 21:22
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Kudos [?]: 282 [0], given: 0

SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1852

Kudos [?]: 2712 [0], given: 193

Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: If x is an integer, then x(x – 1)(x – k) must be evenly divi [#permalink]

### Show Tags

07 Mar 2014, 04:52
Plugged in value of x = 8

It comes up 8 x 7 x (8-k)

Checking for each option available

-4 >>> 8+4 = 12.. Divisible by 3

-2 >>> 8+2 = 10.. Not divisible by 3

-1 >>> 8+1 = 9 .. Divisible by 3

2 >>> 8-2 = 6 .. Divisible by 3

5 >>> 8-5 = 3 .. Divisible by 3

_________________

Kindly press "+1 Kudos" to appreciate

Kudos [?]: 2712 [0], given: 193

Senior Manager
Joined: 20 Dec 2013
Posts: 267

Kudos [?]: 107 [0], given: 29

Location: India
Re: If x is an integer, then x(x – 1)(x – k) must be evenly divi [#permalink]

### Show Tags

08 Mar 2014, 05:51
Hats off to Bunuel for the 30 sec. Approach!Couldn't visualize that solution!

Posted from my mobile device

Kudos [?]: 107 [0], given: 29

Director
Joined: 17 Dec 2012
Posts: 623

Kudos [?]: 534 [0], given: 16

Location: India
Re: If x is an integer, then x(x – 1)(x – k) must be evenly divi [#permalink]

### Show Tags

02 Aug 2014, 01:42
anilnandyala wrote:
If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT

A. -4
B. -2
C. -1
D. 2
E. 5

Since x(x-1)(x-k) is divisible by 3, take a case when x(x-1) is not divisible by 3 and so (x-k) has to be divisible by 3.
Let us take x=8 and x-1=7. Only for the second option we do not get x-k divisible by 3.
_________________

Srinivasan Vaidyaraman
Sravna
http://www.sravnatestprep.com/regularcourse.php

Standardized Approaches

Kudos [?]: 534 [0], given: 16

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7736

Kudos [?]: 17797 [0], given: 235

Location: Pune, India
Re: If x is an integer, then x(x – 1)(x – k) must be evenly divi [#permalink]

### Show Tags

02 Sep 2014, 00:21
VeritasPrepKarishma wrote:
I am providing the theoretical explanation below. Once you get it, you can solve such questions in a few seconds in future!

Notice a few things about integers:
-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16......

Every number is a multiple of 1
Every second number is a multiple of 2
Every third number is a multiple of 3
Every fourth number is a multiple of 4 and so on...

So if I pick any 3 consecutive integers, one and only one of them will be a multiple of 3: e.g. I pick 4, 5, 6 (6 is a multiple of 3) or I pick 11, 12, 13 (12 is a multiple of 3) etc..

x(x - 1)(x - k) will be evenly divisible by 3 if at least one of x, x - 1 and x - k is a multiple of 3. We know from above, (x - 2)(x - 1)x will have a multiple of 3 in it. Also, (x-1)x(x + 1) will have a multiple of 3 in it because they both are products of 3 consecutive integers. So k can be 2 or -1. Eliminate these options.
Now let me write down consecutive integers around x:

(x-5), (x - 4), (x - 3), (x - 2), (x - 1), x, (x + 1), (x + 2), (x + 3), (x + 4), (x + 5) etc

(x - 2)(x - 1)x will have a multiple of 3 in it. x could be the multiple of 3, (x - 1) could be the multiple of 3 or (x - 2) could be the multiple of 3, in which case (x - 5) will also be a multiple of 3.
So in any case, (x - 5)(x - 1)x will have a multiple of 3 in it. So k can be 5.

Similarly, (x-1)x(x + 1) will have a multiple of 3 in it. x could be the multiple of 3, (x - 1) could be the multiple of 3 or (x + 1) could be the multiple of 3, in which case (x + 4) will also be a multiple of 3.
So in any case, (x - 1)x(x + 4) will have a multiple of 3 in it. So k can be -4.

We cannot say whether (x-1)x(x + 2) will have a multiple of 3 in it and hence if k = -2, we cannot say whether the product is evenly divisible by 3.

Quote:
Plz Could you please explain how x-5 will also be a multiple of 3. I couldnot understand that part.

If (x - 2) is a multiple of 3, (x - 5), a number 3 places away from (x - 5) will also be divisible by 3.

Say (x - 2) = 9 (a multiple of 3)
then (x - 5) = 6 (previous multiple of 3)
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Kudos [?]: 17797 [0], given: 235

Manager
Joined: 06 Aug 2013
Posts: 91

Kudos [?]: 3 [0], given: 17

Re: If x is an integer, then x(x – 1)(x – k) must be evenly divi [#permalink]

### Show Tags

03 Oct 2014, 08:45
Bunuel wrote:
anilnandyala wrote:
If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT

-4
-2
-1
2
5

We have the product of 3 integers: (x-1)x(x-k).

Note that the product of 3 integers is divisible by 3 if at least one multiple is divisible by 3. Now, to guarantee that at least one integer out of x, (x – 1), and (x – k) is divisible by 3 these numbers must have different remainders upon division by 3, meaning that one of them should have remainder of 1, another reminder of 2 and the last one remainder of 0, so be divisible by 3.

Next, if k=-2 then we'll have (x-1)x(x+2)=(x-1)x(x-1+3) --> which means that (x-1) and (x+2) will have the same remainder upon division by 3. Thus for k=-2 we won't be sure whether (x-1)x(x-k) is divisible by 3.

30 second approach: 4 out of 5 values of k from answer choices must guarantee divisibility of some expression by 3. Now, these 4 values of k in answer choices must have some pattern: if we get rid of -2 then -4, -1, 2, and 5 creating arithmetic progression with common difference of 3, so -2 is clearly doesn't belong to this pattern.

Hope it helps.

Hi Bunuel,
does "evenly divisible" mean that the dividend on being divided by 3, leave a quotient that is even??
please correct me if i am wrong.

Kudos [?]: 3 [0], given: 17

Math Expert
Joined: 02 Sep 2009
Posts: 42257

Kudos [?]: 132703 [0], given: 12335

Re: If x is an integer, then x(x – 1)(x – k) must be evenly divi [#permalink]

### Show Tags

03 Oct 2014, 08:49
arnabs wrote:
Bunuel wrote:
anilnandyala wrote:
If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT

-4
-2
-1
2
5

We have the product of 3 integers: (x-1)x(x-k).

Note that the product of 3 integers is divisible by 3 if at least one multiple is divisible by 3. Now, to guarantee that at least one integer out of x, (x – 1), and (x – k) is divisible by 3 these numbers must have different remainders upon division by 3, meaning that one of them should have remainder of 1, another reminder of 2 and the last one remainder of 0, so be divisible by 3.

Next, if k=-2 then we'll have (x-1)x(x+2)=(x-1)x(x-1+3) --> which means that (x-1) and (x+2) will have the same remainder upon division by 3. Thus for k=-2 we won't be sure whether (x-1)x(x-k) is divisible by 3.

30 second approach: 4 out of 5 values of k from answer choices must guarantee divisibility of some expression by 3. Now, these 4 values of k in answer choices must have some pattern: if we get rid of -2 then -4, -1, 2, and 5 creating arithmetic progression with common difference of 3, so -2 is clearly doesn't belong to this pattern.

Hope it helps.

Hi Bunuel,
does "evenly divisible" mean that the dividend on being divided by 3, leave a quotient that is even??
please correct me if i am wrong.

No, evenly divisible means divisible without remainder, so simply divisible.
_________________

Kudos [?]: 132703 [0], given: 12335

Manager
Joined: 06 Aug 2013
Posts: 91

Kudos [?]: 3 [0], given: 17

Re: If x is an integer, then x(x – 1)(x – k) must be evenly divi [#permalink]

### Show Tags

03 Oct 2014, 08:56
Bunuel wrote:
anilnandyala wrote:
If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT

-4
-2
-1
2
5

We have the product of 3 integers: (x-1)x(x-k).

Note that the product of 3 integers is divisible by 3 if at least one multiple is divisible by 3. Now, to guarantee that at least one integer out of x, (x – 1), and (x – k) is divisible by 3 these numbers must have different remainders upon division by 3, meaning that one of them should have remainder of 1, another reminder of 2 and the last one remainder of 0, so be divisible by 3.

Next, if k=-2 then we'll have (x-1)x(x+2)=(x-1)x(x-1+3) --> which means that (x-1) and (x+2) will have the same remainder upon division by 3. Thus for k=-2 we won't be sure whether (x-1)x(x-k) is divisible by 3.

30 second approach: 4 out of 5 values of k from answer choices must guarantee divisibility of some expression by 3. Now, these 4 values of k in answer choices must have some pattern: if we get rid of -2 then -4, -1, 2, and 5 creating arithmetic progression with common difference of 3, so -2 is clearly doesn't belong to this pattern.

Hope it helps.

I am sorry but i did not really get the solution. A little more elaboration would help Bunuel. My main concern here is, if x(x-1)(x-k) were to be evenly divisible, then plugging any value for x(lets say 3) should make it evenly divisible by 3.

Kudos [?]: 3 [0], given: 17

Math Expert
Joined: 02 Sep 2009
Posts: 42257

Kudos [?]: 132703 [0], given: 12335

Re: If x is an integer, then x(x – 1)(x – k) must be evenly divi [#permalink]

### Show Tags

03 Oct 2014, 09:59
arnabs wrote:
Bunuel wrote:
anilnandyala wrote:
If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT

-4
-2
-1
2
5

We have the product of 3 integers: (x-1)x(x-k).

Note that the product of 3 integers is divisible by 3 if at least one multiple is divisible by 3. Now, to guarantee that at least one integer out of x, (x – 1), and (x – k) is divisible by 3 these numbers must have different remainders upon division by 3, meaning that one of them should have remainder of 1, another reminder of 2 and the last one remainder of 0, so be divisible by 3.

Next, if k=-2 then we'll have (x-1)x(x+2)=(x-1)x(x-1+3) --> which means that (x-1) and (x+2) will have the same remainder upon division by 3. Thus for k=-2 we won't be sure whether (x-1)x(x-k) is divisible by 3.

30 second approach: 4 out of 5 values of k from answer choices must guarantee divisibility of some expression by 3. Now, these 4 values of k in answer choices must have some pattern: if we get rid of -2 then -4, -1, 2, and 5 creating arithmetic progression with common difference of 3, so -2 is clearly doesn't belong to this pattern.

Hope it helps.

I am sorry but i did not really get the solution. A little more elaboration would help Bunuel. My main concern here is, if x(x-1)(x-k) were to be evenly divisible, then plugging any value for x(lets say 3) should make it evenly divisible by 3.

Have you checked this: if-x-is-an-integer-then-x-x-1-x-k-must-be-evenly-divi-106310.html#p846137 ?
_________________

Kudos [?]: 132703 [0], given: 12335

Manager
Joined: 06 Aug 2013
Posts: 91

Kudos [?]: 3 [0], given: 17

Re: If x is an integer, then x(x – 1)(x – k) must be evenly divi [#permalink]

### Show Tags

03 Oct 2014, 11:17
Bunuel wrote:
vjsharma25 wrote:
Hi bunuel,
I don't understand the problem language, it says

If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT

how does it matter whats the value of K, i can choose x = 3 and the expression will always be divisible by 3.

Am i missing any minor yet important point?

Stem says: "If x is an integer, then x(x – 1)(x – k) must be evenly divisible by three when k is any of the following values EXCEPT"

The important word in the stem is "MUST", which means that we should guarantee the divisibility by 3 no matter the value of x (for ANY integer value of x), so you can not arbitrary pick its value.

Hope it's clear.

that was so helpful bunuel ,thank you so much!!!

Kudos [?]: 3 [0], given: 17

Re: If x is an integer, then x(x – 1)(x – k) must be evenly divi   [#permalink] 03 Oct 2014, 11:17

Go to page    1   2    Next  [ 28 posts ]

Display posts from previous: Sort by