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

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

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17 Oct 2009, 14:15
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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, 02:35, edited 1 time in total.

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17 Oct 2009, 14:32
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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.

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19 Oct 2009, 23:18
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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.

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..
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02 Dec 2010, 11:16
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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
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17334 [10], given: 232 Retired Moderator Status: 2000 posts! I don't know whether I should feel great or sad about it! LOL Joined: 04 Oct 2009 Posts: 1637 Kudos [?]: 1104 [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 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/my-ir-logbook-diary-133264.html GMAT Club Premium Membership - big benefits and savings Kudos [?]: 1104 [0], given: 109 Math Expert Joined: 02 Sep 2009 Posts: 41873 Kudos [?]: 128579 [2], given: 12180 Re: positive integers [#permalink] ### Show Tags 12 Dec 2010, 10:48 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 [?]: 128579 [2], given: 12180 Intern Joined: 06 Dec 2010 Posts: 17 Kudos [?]: [0], given: 1 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 Kudos [?]: [0], given: 1 Moderator Joined: 01 Sep 2010 Posts: 3355 Kudos [?]: 9042 [1], given: 1152 Re: positive integers [#permalink] ### Show Tags 26 Dec 2010, 17:37 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 [?]: 9042 [1], given: 1152 Manager Joined: 19 Dec 2010 Posts: 136 Kudos [?]: 32 [0], given: 12 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 Kudos [?]: 32 [0], given: 12 SVP Joined: 16 Nov 2010 Posts: 1599 Kudos [?]: 592 [0], given: 36 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 Kudos [?]: 592 [0], given: 36 Intern Joined: 06 Sep 2010 Posts: 40 Kudos [?]: 7 [0], given: 0 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 !! 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, 10:06 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: 116 Kudos [?]: 148 [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, 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. Kudos [?]: 148 [0], given: 15 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7670 Kudos [?]: 17334 [1], given: 232 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 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

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

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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!

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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....
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03 Aug 2013, 04:26
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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.
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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 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

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

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