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

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

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Updated on: 24 Oct 2012, 04:18
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00:00

Difficulty:

35% (medium)

Question Stats:

67% (02:05) correct 33% (02:24) wrong based on 243 sessions

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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, 03:48.
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]

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24 Oct 2012, 04:17
2
1
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, 05:12
1
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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Re: If x, y, and z are positive integers, x is a factor of 2y  [#permalink]

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

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19 Sep 2015, 01:35
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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17 Jun 2017, 20:15
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.
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Posts: 51214
Re: If x, y, and z are positive integers, x is a factor of 2y  [#permalink]

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18 Jun 2017, 02:07
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, 20:32
Ok.. I get your point. Can you give me an example where E would not be an integer?
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Posts: 51214
Re: If x, y, and z are positive integers, x is a factor of 2y  [#permalink]

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18 Jun 2017, 20:48
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]

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30 Jan 2018, 13:57
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.

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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31 Jan 2018, 03:25
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.

Regards,
Saquib
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Re: If x, y, and z are positive integers, x is a factor of 2y &nbs [#permalink] 31 Jan 2018, 03:25
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