GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 13 Nov 2019, 02:11

Close

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
Your Progress

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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 59003
If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?  [#permalink]

Show Tags

New post 12 Sep 2019, 21:30
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

55% (01:37) correct 45% (01:36) wrong based on 55 sessions

HideShow timer Statistics

Senior Manager
Senior Manager
avatar
P
Joined: 18 May 2019
Posts: 436
GMAT ToolKit User Premium Member CAT Tests
Re: If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?  [#permalink]

Show Tags

New post Updated on: 12 Sep 2019, 22:57
1
We are to determine if x^3 - 3x^2 + 2x divisible by 4, and that x is a positive integer.

X^3 - 3x^2 + 2x = x(x-2)(x-1)

From 1, we know x=4y+4 where y is an integer. This statement is sufficient because for every value of y, x is an integral multiple of 4. Hence would be divisible by 4.

From 2, x=2z+2. This also sufficient because we know that x is either 2, a multiple of 4, or 4r+2 where r is an integer. When x=2, we get the function reduced to 0, which is divisible by 4. When x=4r+2, the term (x-2) will still make it a multiple of 4 hence it will be divisible by 4.

The answer is D in my view.

Originally posted by eakabuah on 12 Sep 2019, 21:52.
Last edited by eakabuah on 12 Sep 2019, 22:57, edited 1 time in total.
Senior Manager
Senior Manager
avatar
G
Joined: 01 Mar 2019
Posts: 261
Location: India
Concentration: Strategy, Social Entrepreneurship
Schools: Ross '22, ISB '20, NUS '20
GPA: 4
Reviews Badge
Re: If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?  [#permalink]

Show Tags

New post 12 Sep 2019, 22:15
1
x3−3x2+2x divisible by 4?

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

(4y+4)^3-3(4y+4)^2+2(4y+4)
=4(a) = result will be a multiple of 4

(2) x=2z+2, where z is an integer

(2z+2)^3-3(2z+2)^2+2(2z+2)
= 4(b) = result will be a multiple of 4

SO OA:D, both are sufficient individually.
Senior Manager
Senior Manager
User avatar
P
Status: Whatever it takes!
Joined: 10 Oct 2018
Posts: 383
GPA: 4
Re: If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?  [#permalink]

Show Tags

New post Updated on: 13 Sep 2019, 22:35
1
If x is a positive integer, is \(x^3\)−3\(x^2\)+2x divisible by 4?
\(\frac{x(x^2-3x+2)}{4}\)=?

\(\frac{x(x-1)(x-2)}{4}\)=?

(1) x=4y+4, where y is an integer
If y=0, x=4 then \(\frac{4(4-1)(4-2)}{4}\)=?.........YES
If y=1, x=8 then \(\frac{8(8-1)(8-2)}{4}\)=?.........YES
If y=5, x=24 then \(\frac{24(24-1)(24-2)}{4}\)=?.........YES
Always yes. SUFFICIENT!

(2) x=2z+2, where z is an integer
If z=1, x=4 then \(\frac{4(4-1)(4-2)}{4}\)=?.........YES
If z=2, x=6 then \(\frac{6(6-1)(6-2)}{4}\)=?.........YES
Always yes. SUFFICIENT!

IMO answer is option D

Posted from my mobile device
_________________
Kudos OK Please!!

ALL ABOUT GMAT- \(https://exampal.com/gmat/blog/gmat-score-explained\)

Originally posted by EncounterGMAT on 12 Sep 2019, 22:51.
Last edited by EncounterGMAT on 13 Sep 2019, 22:35, edited 1 time in total.
Manager
Manager
avatar
S
Joined: 23 Nov 2018
Posts: 229
GMAT 1: 650 Q49 V28
GPA: 4
Reviews Badge CAT Tests
Re: If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?  [#permalink]

Show Tags

New post 12 Sep 2019, 23:07
its always important to simplify the questions statement; we get

is x(x-1)(x-2) divisible by 4?----1

a) x=4y+4 => 4(y+1)
when we substitute into 1 we get 4(y+1)(x-1)(x-2) the 4 taken out common is divisible therefore yes! a is sufficient

b)x= 2z+2 => 2(z+1)
substituting we get
2(z+1)(2z+1)(2z)
=> 4*z*(z+1)*(2z+1).... yes b is sufficient

therefore "D" is the answer
_________________
.
Senior Manager
Senior Manager
User avatar
P
Joined: 16 Jan 2019
Posts: 498
Location: India
Concentration: General Management
WE: Sales (Other)
Re: If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?  [#permalink]

Show Tags

New post 12 Sep 2019, 23:17
\(x^3-3x^2+2x=x(x^2-3x+2)=x(x-1)(x-2)\)

\(x(x-1)(x-2)\) is the product of three consecutive integers. This product will surely be divisible by \(4\) when two of three consecutive integers are even

This can only be possible when \(x\) and \((x-2)\) are even.

So all we need to know is whether \(x\) is even

Statements (1) and (2) each independently state that \(x\) is even (Because even+even=even). So we have our answer

(1) and (2) are each independently sufficient

Answer is (D)

Posted from my mobile device
GMAT Club Legend
GMAT Club Legend
User avatar
D
Joined: 18 Aug 2017
Posts: 5261
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
GMAT ToolKit User Premium Member CAT Tests
Re: If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?  [#permalink]

Show Tags

New post 13 Sep 2019, 01:44
given eqn
x^3−3x^2+2x can be re written as x(x^2-3z+2)
#1 x=4y+4, where y is an integer
test with y=even and odd integer ; value of x will always be a multiple of 4 since x=4(y+1) sufficient
#2
x=2z+2, where z is an integer
test with z=1,z=2 we get even integer value of x
which is >2 ; hence x(x^2-3z+2) will be divisible by 4 ; 2^2
sufficient
IMO D



If x is a positive integer, is x^3−3x^2+2x divisible by 4?

(1) x=4y+4, where y is an integer
(2)x=2z+2, where z is an integer
Senior Manager
Senior Manager
avatar
G
Joined: 07 Mar 2019
Posts: 372
Location: India
GMAT 1: 580 Q43 V27
WE: Sales (Energy and Utilities)
CAT Tests
Re: If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?  [#permalink]

Show Tags

New post 13 Sep 2019, 02:47
1
If x is a positive integer, is \(x^3 − 3x^2 + 2x\) divisible by 4?

\(x^3 − 3x^2 + 2x = x(x - 2)(x - 1)\)

So if x is a multiple of 4 then \(x^3 − 3x^2 + 2x\) is divisible by 4.

(1) \(x = 4y + 4\), where y is an integer
Since \(x = 4(y + 1)\) where y ≥ 0 then \(x^3 − 3x^2 + 2x\) is divisible by 4 always.

SUFFICIENT.

(2) \(x = 2z + 2\), where z is an integer
\(x = 2(z + 1)\) where y ≥ 0

If z is odd then x = 4k where k > 0 integer, then \(x^3 − 3x^2 + 2x\) is divisible by 4 always
Or if z is even then x = 6, 10, 14 then \(x^3 − 3x^2 + 2x\) is not divisible by 4.

INSUFFICIENT.

Answer (A).
_________________
Ephemeral Epiphany..!

GMATPREP1 590(Q48,V23) March 6, 2019
GMATPREP2 610(Q44,V29) June 10, 2019
GMATPREPSoft1 680(Q48,V35) June 26, 2019
Director
Director
avatar
P
Joined: 24 Nov 2016
Posts: 748
Location: United States
Re: If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?  [#permalink]

Show Tags

New post 13 Sep 2019, 03:34
Quote:
If x is a positive integer, is \(x^3−3x^2+2x\) divisible by 4?

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


\(x^3−3x^2+2x…x(x^2-3x+2)\)

(1) x=4y+4, where y is an integer: sufic.
\(x=4(y+1)…x=multiple.4…\frac{m4(x^2-3x+2)}{4}=x^2-3x+2=integer\)

(2) x=2z+2, where z is an integer: sufic.
\(x=2z+2…x(x^2-3x+2)=(2z+2)((2z+2)^2-3(2z+2)+2)=(2z+2)(4z^2+4+8z-6z-6+2);\)
\(…=(2z+2)(4z^2+2z)=8z^3+4z^2+8z^2+4z=8z^3+12z^2+4z=4(2z^3+3z^2+z);\)
\(…=\frac{4(2z^3+3z^2+z)}{4}=(2z^3+3z^2+z)=integer\)

Answer (D)
Senior Manager
Senior Manager
User avatar
G
Joined: 10 Aug 2018
Posts: 338
Location: India
Concentration: Strategy, Operations
WE: Operations (Energy and Utilities)
Reviews Badge CAT Tests
Re: If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?  [#permalink]

Show Tags

New post 13 Sep 2019, 04:38
IMO it's D
(1) x=4y+4
(2) x=2z+2
Both are sufficient.

a) Because x^3 - 3x^2 + 2x could be written as x*(x^2-3x+2)
b) from both statements we know that x is even so it has a 2 in it.
c) Similarly (x^2-3x+2) would be even and have a 2 in it.
d) The total term would be divisible by 4(2*2)
_________________
On the way to get into the B-school and I will not leave it until I win. WHATEVER IT TAKES.

" I CAN AND I WILL"

GMAT:[640 Q44, V34, IR4, AWA5]
Intern
Intern
avatar
B
Joined: 04 Sep 2019
Posts: 7
Re: If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?  [#permalink]

Show Tags

New post 13 Sep 2019, 08:48
1
If x is a positive integer, is \(x^3−3x^2+2x\) divisible by 4?

(1) x=4y+4, where y is an integer
x = 4(y+1) and the expression is a multiple of x, hence it is also a multiple of 4.
Sufficient.

(2) x=2z+2, where z is an integer
x = 2(z+1)
Evaluating 3 parts of the expression separately:
\(x^3 = 2^3 (z+1)^3\) multiple of 8
\(3x^2 = 3*2^2*(z+1)^2\) multiple of 4
2x = 2*2 multiple of 4
Sufficient.

Both 1 and 2 can independently answer the question.
So D.
GMAT Club Bot
Re: If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?   [#permalink] 13 Sep 2019, 08:48
Display posts from previous: Sort by

If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne