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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]

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New post Updated on: 24 Oct 2012, 05:18
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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) \(\frac{zy}{3x}\)

Originally posted by LM on 24 Oct 2012, 04:48.
Last edited by Bunuel on 24 Oct 2012, 05:18, 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]

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New post 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 \(x\): \(\frac{z}{3y}*x=integer*x\) --> \(\frac{zx}{3y}=x*integer=integer*integer=integer\).

Answer: D.

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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New post 31 Oct 2012, 06:12
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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 \(x\): \(\frac{z}{3y}*x=integer*x\) --> \(\frac{zx}{3y}=x*integer=integer*integer=integer\).

Answer: D.

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

Please confirm
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Re: If x, y, and z are positive integers, x is a factor of 2y  [#permalink]

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New post 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 \(x\): \(\frac{z}{3y}*x=integer*x\) --> \(\frac{zx}{3y}=x*integer=integer*integer=integer\).

Answer: D.

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

Please confirm


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]

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New post 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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Re: If x, y, and z are positive integers, x is a factor of 2y  [#permalink]

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New post 17 Jun 2017, 21:15
I have a doubt here. Option E is also possible.
The answer option is
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.
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Re: If x, y, and z are positive integers, x is a factor of 2y  [#permalink]

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New post 18 Jun 2017, 03:07
laasshetty wrote:
I have a doubt here. Option E is also possible.
The answer option is
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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New post 18 Jun 2017, 21:32
Ok.. I get your point. Can you give me an example where E would not 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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New post 18 Jun 2017, 21:48
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Re: If x, y, and z are positive integers, x is a factor of 2y  [#permalink]

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New post 30 Jan 2018, 14:57
Hi All,

I'm a big fan of TESTing VALUES (which is showcased in several of the explanations) - it's perfect for these types of questions. In addition, the answer choices are written in such a way that you can answer this question using Number Property rules....

In the prompt, we're told that "3Y is a factor of Z." This is just a wordy way of saying Z/3Y is an INTEGER. We're also told that X, Y and Z are positive integers.

Looking at the answers, what do you notice about answer D?

ZX/3Y....

It's interesting that the letters in in the numerator are NOT in alphabetical order. I wonder why THAT is? Is it to hide the fact that this answer has Z/3Y in it? We already know that Z/3Y is an integer and we know that X is an integer too.

Answer D literally translates into (integer)(integer)....which is ALWAYS 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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New post 31 Jan 2018, 04:25
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) \(\frac{zy}{3x}\)


Solution



    • Given: x is factor of 2y.

      o Thus, we can write \(2y = k * x\), where k is an integer

      o \(\frac{x}{y} = \frac{2}{k}\)

    • Given: 3y is a factor of z.

      o Thus, we can write \(z = q * 3y\), where q is an integer

      o \(\frac{y}{z} = \frac{1}{3q}\)

    • Thus \(x : y : z = 2: k: 3qk\)

Now let us consider the options one by one and check which of the following MUST be an integer

    1. \(\frac{y}{x} = \frac{k}{2}\)
      a. We cannot be sure, since we do not know the value of k

    2. \(\frac{2y}{6} = \frac{k}{3}\)
      a. Again, we cannot be sure, since we do not know the value of k

    3. \(\frac{xy}{3z} = \frac{2k}{9qk} = \frac{2}{9q}\)
      a. Again, we cannot be sure, since we do not know the value of q

    4. \(\frac{zx}{3y} = \frac{3qk*2}{3k} = 2q\)
      a. This has to be an integer, since 2q is an integer

    At this point, we can stop and mark the answer as D, but just to be sure, let’s check the last option.

    5. \(\frac{zy}{3x} = \frac{3qk*k}{3*2}= \frac{qk^2}{2}\)
      a. We cannot be sure this is an integer, since we do not know the value of k or q.

Correct Answer: Option D

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Re: If x, y, and z are positive integers, x is a factor of 2y  [#permalink]

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Re: If x, y, and z are positive integers, x is a factor of 2y   [#permalink] 16 Feb 2019, 04:00
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