It is currently 13 Dec 2017, 10:59

# Decision(s) Day!:

CHAT Rooms | Ross R1 | Kellogg R1 | Darden R1 | Tepper R1

### 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

# The positive integers x, y, and z are such that x is a

Author Message
TAGS:

### Hide Tags

Intern
Joined: 19 Jun 2009
Posts: 28

Kudos [?]: 129 [1], given: 1

The positive integers x, y, and z are such that x is a [#permalink]

### Show Tags

17 Oct 2009, 13:15
1
KUDOS
15
This post was
BOOKMARKED
00:00

Difficulty:

55% (hard)

Question Stats:

61% (01:01) correct 39% (01:06) wrong based on 900 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.
[Reveal] Spoiler: OA

Last edited by Bunuel on 25 Feb 2012, 01:35, edited 1 time in total.

Kudos [?]: 129 [1], given: 1

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7792

Kudos [?]: 18112 [10], given: 236

Location: Pune, India

### Show Tags

02 Dec 2010, 10:16
10
KUDOS
Expert's post
2
This post was
BOOKMARKED
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.

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
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 18112 [10], given: 236 Math Expert Joined: 02 Sep 2009 Posts: 42583 Kudos [?]: 135518 [7], given: 12697 Re: positive integers [#permalink] ### Show Tags 17 Oct 2009, 13:32 7 This post received KUDOS Expert's post 17 This post was BOOKMARKED 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 non-zero integer $$m$$; y is a factor of z --> $$z=ny$$, for some non-zero 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. _________________ Kudos [?]: 135518 [7], given: 12697 Math Expert Joined: 02 Sep 2009 Posts: 42583 Kudos [?]: 135518 [2], given: 12697 Re: positive integers [#permalink] ### Show Tags 12 Dec 2010, 09:48 2 This post received KUDOS Expert's post metallicafan wrote: Bunuel, you always try to solve the questions algebraically, don't you? Not at all. There are certain GMAT questions which are pretty much only solvable with plug-in or trial and error methods (well at leas in 2-3 minutes). Also many questions can be solved with logic and common sense much quicker than with algebraic approach. So you shouldn't always rely on algebra. Having said that I must add that there are of course other types of questions which are perfect for algebraic approach, plus I often use algebra just to explain a solution. _________________ Kudos [?]: 135518 [2], given: 12697 SVP Joined: 29 Aug 2007 Posts: 2470 Kudos [?]: 867 [1], given: 19 Re: positive integers [#permalink] ### Show Tags 19 Oct 2009, 22:18 1 This post received KUDOS 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/new-to-the-verbal-forum-please-read-this-first-77546.html Math: http://gmatclub.com/forum/new-to-the-math-forum-please-read-this-first-77764.html Gmat: http://gmatclub.com/forum/everything-you-need-to-prepare-for-the-gmat-revised-77983.html GT Kudos [?]: 867 [1], given: 19 Board of Directors Joined: 01 Sep 2010 Posts: 3422 Kudos [?]: 9491 [1], given: 1203 Re: positive integers [#permalink] ### Show Tags 26 Dec 2010, 16:37 1 This post received KUDOS 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 _________________ Kudos [?]: 9491 [1], given: 1203 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7792 Kudos [?]: 18112 [1], given: 236 Location: Pune, India Re: The positive integers x, y, and z are such that x is a [#permalink] ### Show Tags 27 Feb 2012, 04:20 1 This post received KUDOS Expert's post 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 My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Kudos [?]: 18112 [1], given: 236

Verbal Forum Moderator
Joined: 10 Oct 2012
Posts: 627

Kudos [?]: 1406 [1], given: 136

### Show Tags

03 Aug 2013, 03:26
1
KUDOS
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.
_________________

Kudos [?]: 1406 [1], given: 136

Retired Moderator
Status: 2000 posts! I don't know whether I should feel great or sad about it! LOL
Joined: 04 Oct 2009
Posts: 1626

Kudos [?]: 1138 [0], given: 109

Location: Peru
Schools: Harvard, Stanford, Wharton, MIT & HKS (Government)
WE 1: Economic research
WE 2: Banking
WE 3: Government: Foreign Trade and SMEs

### Show Tags

12 Dec 2010, 09:22
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/my-ir-logbook-diary-133264.html

GMAT Club Premium Membership - big benefits and savings

Kudos [?]: 1138 [0], given: 109

Intern
Joined: 06 Dec 2010
Posts: 16

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

### Show Tags

12 Dec 2010, 18:36
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

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

Manager
Joined: 19 Dec 2010
Posts: 133

Kudos [?]: 32 [0], given: 12

### Show Tags

17 Mar 2011, 22:07
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

Kudos [?]: 32 [0], given: 12

TOEFL Forum Moderator
Joined: 16 Nov 2010
Posts: 1586

Kudos [?]: 607 [0], given: 40

Location: United States (IN)
Concentration: Strategy, Technology

### Show Tags

17 Mar 2011, 22:52
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.

_________________

Formula of Life -> Achievement/Potential = k * Happiness (where k is a constant)

GMAT Club Premium Membership - big benefits and savings

Kudos [?]: 607 [0], given: 40

Intern
Joined: 06 Sep 2010
Posts: 39

Kudos [?]: 7 [0], given: 0

### Show Tags

15 Apr 2011, 04:24
At first, mistook "factor" for "multiple" came with answer E.
Later, understood that the problem was so easy...just plug and play !!

Kudos [?]: 7 [0], given: 0

Manager
Joined: 16 May 2011
Posts: 71

Kudos [?]: 18 [0], given: 2

Re: The positive integers x, y, and z are such that x is a [#permalink]

### Show Tags

24 Feb 2012, 09:06
if we plug and play, why can't we test X = 1? then y/n..

Kudos [?]: 18 [0], given: 2

Manager
Joined: 21 Feb 2012
Posts: 115

Kudos [?]: 153 [0], given: 15

Location: India
Concentration: Finance, General Management
GMAT 1: 600 Q49 V23
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, 09:46
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.

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.

Kudos [?]: 153 [0], given: 15

Senior Manager
Joined: 13 Aug 2012
Posts: 457

Kudos [?]: 570 [0], given: 11

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, 21:49
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!

_________________

Impossible is nothing to God.

Kudos [?]: 570 [0], given: 11

Director
Joined: 29 Nov 2012
Posts: 863

Kudos [?]: 1486 [0], given: 543

### Show Tags

03 Aug 2013, 01:04
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/gmat-prep-software-analysis-and-what-if-scenarios-146146.html

Kudos [?]: 1486 [0], given: 543

Current Student
Joined: 06 Sep 2013
Posts: 1965

Kudos [?]: 759 [0], given: 355

Concentration: Finance

### Show Tags

10 Oct 2013, 14:01
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 non-zero integer $$m$$;
y is a factor of z --> $$z=ny$$, for some non-zero 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.

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

Hope it helps

Kudos [?]: 759 [0], given: 355

Retired Moderator
Joined: 12 Aug 2015
Posts: 2209

Kudos [?]: 901 [0], given: 607

GRE 1: 323 Q169 V154
Re: The positive integers x, y, and z are such that x is a [#permalink]

### Show Tags

21 Aug 2016, 08:10
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
_________________

Give me a hell yeah ...!!!!!

Kudos [?]: 901 [0], given: 607

Non-Human User
Joined: 09 Sep 2013
Posts: 14868

Kudos [?]: 287 [0], given: 0

Re: The positive integers x, y, and z are such that x is a [#permalink]

### Show Tags

26 Oct 2017, 08:43
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 [?]: 287 [0], given: 0

Re: The positive integers x, y, and z are such that x is a   [#permalink] 26 Oct 2017, 08:43
Display posts from previous: Sort by