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# If x, y, and z are positive integers, x is a factor of 2y

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Director
Joined: 03 Sep 2006
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If x, y, and z are positive integers, x is a factor of 2y [#permalink]

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24 Oct 2012, 04:48
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If x, y, and z are positive integers, x is a factor of 2y, and 3y is a factor of z, which of the following must also be an integer?

A) $$\frac{y}{x}$$

B) $$\frac{2y}{6}$$

C) $$\frac{xy}{3z}$$

D) $$\frac{zx}{3y}$$

E) $$\frac{zy}{3x}$$
[Reveal] Spoiler: OA

Last edited by Bunuel on 24 Oct 2012, 05:18, edited 2 times in total.
Renamed the topic and edited the question.
Math Expert
Joined: 02 Sep 2009
Posts: 39662
Re: If x, y, and z are positive integers, x is a factor of 2y [#permalink]

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24 Oct 2012, 05:17
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LM wrote:
If x, y, and z are positive integers, x is a factor of 2y, and 3y is a factor of z, which of the following must also be an integer?

A) $$\frac{y}{x}$$

B) $$\frac{2y}{6}$$

C) $$\frac{xy}{3z}$$

D) $$\frac{zx}{3y}$$

E) $$\frac{zy}{3x}$$

$$x$$ is a factor of $$2y$$, means that $$\frac{2y}{x}=integer$$.

Similarly, $$3y$$ is a factor of $$z$$, means that $$\frac{z}{3y}=integer$$. Multiply both sides of this equation by integer $$x$$: $$\frac{z}{3y}*x=integer*x$$ --> $$\frac{zx}{3y}=x*integer=integer*integer=integer$$.

Hope it's clear.
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Re: If x, y, and z are positive integers, x is a factor of 2y [#permalink]

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31 Oct 2012, 06:12
Bunuel wrote:
LM wrote:
If x, y, and z are positive integers, x is a factor of 2y, and 3y is a factor of z, which of the following must also be an integer?

A) $$\frac{y}{x}$$

B) $$\frac{2y}{6}$$

C) $$\frac{xy}{3z}$$

D) $$\frac{zx}{3y}$$

E) $$\frac{zy}{3x}$$

$$x$$ is a factor of $$2y$$, means that $$\frac{2y}{x}=integer$$.

Similarly, $$3y$$ is a factor of $$z$$, means that $$\frac{z}{3y}=integer$$. Multiply both sides of this equation by integer $$x$$: $$\frac{z}{3y}*x=integer*x$$ --> $$\frac{zx}{3y}=x*integer=integer*integer=integer$$.

Hope it's clear.

Hello Bunuel,

Can we simplify the options and then attempt the Questions

For Option 5, ZY/3X can be simplified as 3Y*Y/3X --> Y*Y/X which may or may not be true.
For example looking at option 4, ZX/3Y, if simplify it as ---> Z=3y and then the eqn becomes 3y*y/ 3y which gives x only

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Joined: 02 Sep 2009
Posts: 39662
Re: If x, y, and z are positive integers, x is a factor of 2y [#permalink]

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01 Nov 2012, 07:13
mridulparashar1 wrote:
Bunuel wrote:
LM wrote:
If x, y, and z are positive integers, x is a factor of 2y, and 3y is a factor of z, which of the following must also be an integer?

A) $$\frac{y}{x}$$

B) $$\frac{2y}{6}$$

C) $$\frac{xy}{3z}$$

D) $$\frac{zx}{3y}$$

E) $$\frac{zy}{3x}$$

$$x$$ is a factor of $$2y$$, means that $$\frac{2y}{x}=integer$$.

Similarly, $$3y$$ is a factor of $$z$$, means that $$\frac{z}{3y}=integer$$. Multiply both sides of this equation by integer $$x$$: $$\frac{z}{3y}*x=integer*x$$ --> $$\frac{zx}{3y}=x*integer=integer*integer=integer$$.

Hope it's clear.

Hello Bunuel,

Can we simplify the options and then attempt the Questions

For Option 5, ZY/3X can be simplified as 3Y*Y/3X --> Y*Y/X which may or may not be true.
For example looking at option 4, ZX/3Y, if simplify it as ---> Z=3y and then the eqn becomes 3y*y/ 3y which gives x only

We are not given that z=3y, we are given that z=3y*integer (3y is a factor of z). Now, if we substitute z in $$\frac{zx}{3y}$$, we'l get: $$\frac{zx}{3y}=\frac{(3y*integer)*x}{3y}=integer*x=integer$$.

Hope it helps.
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Posts: 42
Re: If x, y, and z are positive integers, x is a factor of 2y [#permalink]

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19 Sep 2015, 02:35
if we take x=4, y=2 and z=12...
x(=2) is factor of 2y(=4)
and 3y(=6) is a factor of z(=12)......
D it is.......
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Posts: 6
Re: If x, y, and z are positive integers, x is a factor of 2y [#permalink]

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17 Jun 2017, 21:15
I have a doubt here. Option E is also possible.
zy/3x
z is divisible by 3y. that means z/3 will always yield an integer.
y/x will also give an integer.
When we multiply these 2 the result should be an integer as all x,y,z are integers. Please clarify if I am missing something.
Math Expert
Joined: 02 Sep 2009
Posts: 39662
Re: If x, y, and z are positive integers, x is a factor of 2y [#permalink]

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18 Jun 2017, 03:07
laasshetty wrote:
I have a doubt here. Option E is also possible.
zy/3x
z is divisible by 3y. that means z/3 will always yield an integer.
y/x will also give an integer.
When we multiply these 2 the result should be an integer as all x,y,z are integers. Please clarify if I am missing something.

You are not reading the question carefully.

The question asks: which of the following MUST also be an integer? Not COULD also be an integer?
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Re: If x, y, and z are positive integers, x is a factor of 2y [#permalink]

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18 Jun 2017, 21:32
Ok.. I get your point. Can you give me an example where E would not be an integer?
Math Expert
Joined: 02 Sep 2009
Posts: 39662
Re: If x, y, and z are positive integers, x is a factor of 2y [#permalink]

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18 Jun 2017, 21:48
laasshetty wrote:
Ok.. I get your point. Can you give me an example where E would not be an integer?

Yes.

$$x$$ is a factor of $$2y$$, means that $$\frac{2y}{x}=integer$$. Say y = 2 and x = 1.

$$3y$$ is a factor of $$z$$, means that $$\frac{z}{3y}=integer$$. Say z = 3 and y = 1.

E) $$\frac{zy}{3x}=\frac{3*1}{3*2}=\frac{1}{2} \neq integer$$.

Hope it helps.
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Re: If x, y, and z are positive integers, x is a factor of 2y   [#permalink] 18 Jun 2017, 21:48
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