Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 19 Jun 2009
Posts: 27

The positive integers x, y, and z are such that x is a
[#permalink]
Show Tags
Updated on: 25 Feb 2012, 02:35
Question Stats:
61% (01:36) correct 39% (01:50) wrong based on 1057 sessions
HideShow timer Statistics
The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even? (1) xz is even (2) y is even.
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by amitgovin on 17 Oct 2009, 14:15.
Last edited by Bunuel on 25 Feb 2012, 02:35, edited 1 time in total.
Added the OA




Math Expert
Joined: 02 Sep 2009
Posts: 50007

Re: positive integers
[#permalink]
Show Tags
17 Oct 2009, 14:32
amitgovin wrote: The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
(1) xz is even
(2) y is even. Given: x is a factor of y > \(y=mx\), for some nonzero integer \(m\); y is a factor of z > \(z=ny\), for some nonzero integer \(n\); So, \(z=mnx\). Question: is z even? Note that \(z\) will be even if either \(x\) or \(y\) is even (1) \(xz\) even > either \(z\) even, so the answer is directly YES or \(x\) is even (or both). But if \(x\) is even and as \(z=mnx\) then z must be even too (one of the multiples of z is even, so z is even too). Sufficient. (2) \(y\) even > as \(z=ny\) then as one of the multiples of z even > z even. Sufficient. Answer: D.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics




SVP
Joined: 29 Aug 2007
Posts: 2395

Re: positive integers
[#permalink]
Show Tags
19 Oct 2009, 23:18
amitgovin wrote: The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
1) xz is even 2) y is even.
Please explain. thanks. y/x = k where k is an integer. y = xk ....................i z/y = m where m is an integer. z = ym = xkm .....................ii If a factor is even, then the source of the factor must be even. 1) If xz is even, z must be even because x may or may not be an even because x is a factor of z but z must be even. SUFF. 2) If y is even, z must be even because y is a factor of z. SUFF.. D..
_________________
Verbal: http://gmatclub.com/forum/newtotheverbalforumpleasereadthisfirst77546.html Math: http://gmatclub.com/forum/newtothemathforumpleasereadthisfirst77764.html Gmat: http://gmatclub.com/forum/everythingyouneedtoprepareforthegmatrevised77983.html
GT



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8399
Location: Pune, India

Re: positive integers
[#permalink]
Show Tags
02 Dec 2010, 11:16
amitgovin wrote: The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
1) xz is even
2) y is even.
Please explain. thanks. Though Bunuel has provided the solution, I would just like to bring to your notice a train of thought. When we say, "The positive integers x, y, and z are such that x is a factor of y and y is a factor of z.", it implies that if x or y is even, z will be even. e.g. x = 4. Since x is a factor of y, y will be a multiple of 4 and will be even. Since z is a multiple of y, it will also be even. So, in a way, the 2 in x will drive through the entire sequence and make everything even. Once this makes sense to you, it will take 10 secs to arrive at the solution. Stmnt 1: Either x or z (or both which will happen if x is even) is even. In either case, z is even. Stmnt 2: y is even, so z must be even
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
GMAT selfstudy has never been more personalized or more fun. Try ORION Free!



Retired Moderator
Status: 2000 posts! I don't know whether I should feel great or sad about it! LOL
Joined: 04 Oct 2009
Posts: 1211
Location: Peru
Schools: Harvard, Stanford, Wharton, MIT & HKS (Government)
WE 1: Economic research
WE 2: Banking
WE 3: Government: Foreign Trade and SMEs

Re: positive integers
[#permalink]
Show Tags
12 Dec 2010, 10:22
Bunuel, you always try to solve the questions algebraically, don't you?
_________________
"Life’s battle doesn’t always go to stronger or faster men; but sooner or later the man who wins is the one who thinks he can."
My Integrated Reasoning Logbook / Diary: http://gmatclub.com/forum/myirlogbookdiary133264.html
GMAT Club Premium Membership  big benefits and savings



Math Expert
Joined: 02 Sep 2009
Posts: 50007

Re: positive integers
[#permalink]
Show Tags
12 Dec 2010, 10:48



Intern
Joined: 06 Dec 2010
Posts: 16

Re: positive integers
[#permalink]
Show Tags
12 Dec 2010, 19:36
I like to thing of the boxes method. If you draw them out, then x is inside y which is inside z. zx, a 2 will exist inside the box of either z or x (which is itself inside z) so YES y a 2 will exist inside the box of y which is itself z so YES



Board of Directors
Joined: 01 Sep 2010
Posts: 3304

Re: positive integers
[#permalink]
Show Tags
26 Dec 2010, 17:37
VeritasPrepKarishma wrote: amitgovin wrote: The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
1) xz is even
2) y is even.
Please explain. thanks. Though Bunuel has provided the solution, I would just like to bring to your notice a train of thought. When we say, "The positive integers x, y, and z are such that x is a factor of y and y is a factor of z.", it implies that if x or y is even, z will be even. e.g. x = 4. Since x is a factor of y, y will be a multiple of 4 and will be even. Since z is a multiple of y, it will also be even. So, in a way, the 2 in x will drive through the entire sequence and make everything even. Once this makes sense to you, it will take 10 secs to arrive at the solution. Stmnt 1: Either x or z (or both which will happen if x is even) is even. In either case, z is even. Stmnt 2: y is even, so z must be even awesome. your explanations and bunuel as well, are amazing thanks
_________________
COLLECTION OF QUESTIONS AND RESOURCES Quant: 1. ALL GMATPrep questions Quant/Verbal 2. Bunuel Signature Collection  The Next Generation 3. Bunuel Signature Collection ALLINONE WITH SOLUTIONS 4. Veritas Prep Blog PDF Version 5. MGMAT Study Hall Thursdays with Ron Quant Videos Verbal:1. Verbal question bank and directories by Carcass 2. MGMAT Study Hall Thursdays with Ron Verbal Videos 3. Critical Reasoning_Oldy but goldy question banks 4. Sentence Correction_Oldy but goldy question banks 5. Readingcomprehension_Oldy but goldy question banks



Manager
Joined: 19 Dec 2010
Posts: 106

Re: positive integers
[#permalink]
Show Tags
17 Mar 2011, 23:07
Easy if you realize the following: When a is a factor of b AND b is a factor of c THEN a is a factor of c as well.
Hence when either one of these numbers is even, the other has to be even too.. D



Retired Moderator
Joined: 16 Nov 2010
Posts: 1436
Location: United States (IN)
Concentration: Strategy, Technology

Re: positive integers
[#permalink]
Show Tags
17 Mar 2011, 23:52
z = ky y = mx so z = (km)xy (1) > xz is eve means at least x or z is even, and if x = even, then z is also even as it has an even factor. 2 > y is even so z having an even factor is even too. Answer D
_________________
Formula of Life > Achievement/Potential = k * Happiness (where k is a constant)
GMAT Club Premium Membership  big benefits and savings



Intern
Joined: 06 Sep 2010
Posts: 32

Re: positive integers
[#permalink]
Show Tags
15 Apr 2011, 05:24
At first, mistook "factor" for "multiple" came with answer E. Later, understood that the problem was so easy...just plug and play !!



Manager
Joined: 16 May 2011
Posts: 61

Re: The positive integers x, y, and z are such that x is a
[#permalink]
Show Tags
24 Feb 2012, 10:06
if we plug and play, why can't we test X = 1? then y/n..



Manager
Joined: 21 Feb 2012
Posts: 81
Location: India
Concentration: Finance, General Management
GPA: 3.8
WE: Information Technology (Computer Software)

Re: The positive integers x, y, and z are such that x is a
[#permalink]
Show Tags
24 Feb 2012, 10:46
The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even? 1) xz is even 2) y is even. Please explain. thanks.[/quote]
Ans. let us take any 3 numbers, say x=3,y=18,z=54, or x=2,y=4,z=20 1)if xz is even then it means that either x or z is even,say that x is even, now there is no even number which is a factor of odd number, so z is definitely even, now if x is odd as in the above case, still then we can point out that z is even. 2)if y is even then it is clear that z will be even. Thus this question could be answered by any of the two questions.



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8399
Location: Pune, India

Re: The positive integers x, y, and z are such that x is a
[#permalink]
Show Tags
27 Feb 2012, 05:20
dchow23 wrote: if we plug and play, why can't we test X = 1? then y/n.. Plug in method would be far more painful for this question (and most other questions in my opinion). Think how you would go about it: Checking whether stmnt 1 is enough: xz is even If x = 1, y = 1 and z = 2 (so that xz is even), then z is even. If x = 2, y = 2 and z = 4, z is again even. Then you start thinking if you can take some values such that xz is even but z is not... Now you start using logic... Wouldn't you say it is far better to use logic in the first place itself?
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
GMAT selfstudy has never been more personalized or more fun. Try ORION Free!



Senior Manager
Joined: 13 Aug 2012
Posts: 431
Concentration: Marketing, Finance
GPA: 3.23

Re: The positive integers x, y, and z are such that x is a
[#permalink]
Show Tags
22 Jan 2013, 22:49
amitgovin wrote: The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
(1) xz is even (2) y is even.
What is given? \(z = y*N\) \(y = x*R\) 1. \(xz = 2*I\) If x is even, then z is even since \(z = x*R\) If z is even, then z is even. SUFFICIENT! 2. \(y = 2*I\) ==> \(z = 2*I*N\) Definitely EVEN SUFFICIENT! Answer: D
_________________
Impossible is nothing to God.



Director
Joined: 29 Nov 2012
Posts: 775

Re: positive integers
[#permalink]
Show Tags
03 Aug 2013, 02:04
Bunuel wrote: (1) \(xz\) even > either \(z\) even, so the answer is directly YES or \(x\) is even (or both). But if \(x\) is even and as \(z=mnx\) then z must be even too (one of the multiples of z is even, so z is even too). Sufficient.
Can you please provide a numerical example for this part it isn't very clear. Thanks in advance! I thought it was insufficient as there were multiple cases either X or Y even or both....
_________________
Click +1 Kudos if my post helped...
Amazing Free video explanation for all Quant questions from OG 13 and much more http://www.gmatquantum.com/og13th/
GMAT Prep software What if scenarios http://gmatclub.com/forum/gmatprepsoftwareanalysisandwhatifscenarios146146.html



Verbal Forum Moderator
Joined: 10 Oct 2012
Posts: 613

Re: positive integers
[#permalink]
Show Tags
03 Aug 2013, 04:26
fozzzy wrote: Bunuel wrote: (1) \(xz\) even > either \(z\) even, so the answer is directly YES or \(x\) is even (or both). But if \(x\) is even and as \(z=mnx\) then z must be even too (one of the multiples of z is even, so z is even too). Sufficient.
Can you please provide a numerical example for this part it isn't very clear. Thanks in advance! I thought it was insufficient as there were multiple cases either X or Y even or both.... We know that x is a factor of \(y \to y = Ix\) Again, \(z = yI^'\). Now, given that xz  even. Case I:Assume that x = even , z = odd. Now, as x is even, y = even(I can be odd/even,doesn't matter). Again, as y is even, z HAS to be even(\(I^'\) is odd/even, doesn't matter). Thus, if x is even, z IS even. Numerical Example :y = 2*I(x=2). \(z = 6*I^'\). z IS even. Case II : z is even OR (x and z) both are even. Hope this helps.
_________________
All that is equal and notDeep Dive Inequality
Hit and Trial for Integral Solutions



SVP
Joined: 06 Sep 2013
Posts: 1764
Concentration: Finance

Re: positive integers
[#permalink]
Show Tags
10 Oct 2013, 15:01
Bunuel wrote: amitgovin wrote: The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
(1) xz is even
(2) y is even. Given: x is a factor of y > \(y=mx\), for some nonzero integer \(m\); y is a factor of z > \(z=ny\), for some nonzero integer \(n\); So, \(z=mnx\). Question: is z even? Note that \(z\) will be even if either \(x\) or \(y\) is even (1) \(xz\) even > either \(z\) even, so the answer is directly YES or \(x\) is even (or both). But if \(x\) is even and as \(z=mnx\) then z must be even too (one of the multiples of z is even, so z is even too). Sufficient. (2) \(y\) even > as \(z=ny\) then as one of the multiples of z even > z even. Sufficient. Answer: D. Perfect explanation. Remember the factor foundation rule. Also, other properties of factors that might be helpful to have in mind. Just to remind you, The factor foundation rule states that "if a is a factor of b, and b is a factor of c, then a is a factor of c" Also, if 'a' is a factor of 'b', and 'a' is a factor of 'c', then 'a' is a factor of (b+c). In fact, 'a' is a factor of (mb + nc) for all integers 'm' and 'n' If 'a' is a factor of 'b' and 'b' is a factor of 'a', then 'a=b' If 'a' is a factor of 'bc' and gcd (a,b) = 1, then 'a' is a factor of 'c' If 'p' is a prime number and 'p' is a factor of 'ab' then 'p' is a factor of 'a' or 'p' is a factor of 'b' In other words,, any integer is divisible by all of its factors and it is also divisible by all of the factors of its factors Show your appreciation in Kudos. Hope it helps



Current Student
Joined: 12 Aug 2015
Posts: 2638

Re: The positive integers x, y, and z are such that x is a
[#permalink]
Show Tags
21 Aug 2016, 09:10
here is the question stem => y=ax z=by => z=cx for some integers a,b,c now statement 1 => xz is even => cases are => E,E E,O O,E => If x is even then has to be even as z=cx and if x is odd => z has to be even as zx = even => suff statement 2 => y is even => z=by => z has to be even as E*E = E suff SMASH that D
_________________
MBA Financing: INDIAN PUBLIC BANKS vs PRODIGY FINANCE! Getting into HOLLYWOOD with an MBA! The MOST AFFORDABLE MBA programs!STONECOLD's BRUTAL Mock Tests for GMATQuant(700+)AVERAGE GRE Scores At The Top Business Schools!



Director
Joined: 09 Mar 2016
Posts: 942

Re: The positive integers x, y, and z are such that x is a
[#permalink]
Show Tags
15 Mar 2018, 11:50
Bunuel wrote: amitgovin wrote: The positive integers x, y, and z are such that x is a factor of y and y is a factor of z. Is z even?
(1) xz is even
(2) y is even. Given: x is a factor of y > \(y=mx\), for some nonzero integer \(m\); y is a factor of z > \(z=ny\), for some nonzero integer \(n\); So, \(z=mnx\). Question: is z even? Note that \(z\) will be even if either \(x\) or \(y\) is even (1) \(xz\) even > either \(z\) even, so the answer is directly YES or \(x\) is even (or both). But if \(x\) is even and as \(z=mnx\) then z must be even too (one of the multiples of z is even, so z is even too). Sufficient. (2) \(y\) even > as \(z=ny\) then as one of the multiples of z even > z even. Sufficient. Answer: D. Bunuel if x, y, and z are 3, 9 and 27 repectivey then how can A be sufficient ?




Re: The positive integers x, y, and z are such that x is a &nbs
[#permalink]
15 Mar 2018, 11:50



Go to page
1 2
Next
[ 22 posts ]



