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

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If x, y, and z are positive integers, x is a factor of 2y [#permalink]  24 Oct 2012, 03:48
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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, 04:18, edited 2 times in total.
Renamed the topic and edited the question.
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Re: If x, y, and z are positive integers, x is a factor of 2y [#permalink]  24 Oct 2012, 04:17
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Expert's post
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]  31 Oct 2012, 05:12
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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“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”

Math Expert
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Posts: 27505
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Kudos [?]: 42400 [0], given: 6024

Re: If x, y, and z are positive integers, x is a factor of 2y [#permalink]  01 Nov 2012, 06:13
Expert's post
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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Re: If x, y, and z are positive integers, x is a factor of 2y   [#permalink] 01 Nov 2012, 06:13
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# If x, y, and z are positive integers, x is a factor of 2y

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