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Math Expert V
Joined: 02 Sep 2009
Posts: 64314
If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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50 00:00

Difficulty:   25% (medium)

Question Stats: 78% (01:49) correct 22% (01:59) wrong based on 1402 sessions

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If x, y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

(A) (x+z)/z

(B) (y+z)/x

(C) (x+y)/z

(D) (xy)/z

(E) (yz)/x

The Official Guide For GMAT® Quantitative Review, 2ND Edition

Problem Solving
Question: 172
Category: Arithmetic Properties of numbers
Page: 85
Difficulty: 600

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Originally posted by Bunuel on 18 Mar 2014, 00:36.
Last edited by Bunuel on 16 Feb 2019, 03:07, edited 1 time in total.
Updated.
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3394
Re: If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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4
Given
We are given 3 positive integers $$x$$, $$y$$ and $$z$$ such that $$x$$ is a factor of $$y$$ and $$x$$ is a multiple of $$z$$. We are asked to find that among the options given which of them is not necessarily an integer.

Approach
Since $$x$$ is a factor of $$y$$, we can say that $$\frac{y}{x}$$ is an integer...........(1)

Also, as $$x$$ is a multiple of $$z$$ i.e. $$z$$ is a factor of $$x$$, we can say that $$\frac{x}{z}$$ is an integer............(2)

From the above two deductions, we can say that $$\frac{y}{z}$$ will also be an integer as $$x$$ divides $$y$$ completely and $$z$$ divides $$x$$ completely..............(3)

Our endeavor would be to reduce the expressions in the options to one or more of the above 3 forms.

Working Out
(A) $$\frac{x +z}{z} = \frac{x}{z} + 1$$ . Since $$\frac{x}{z}$$ is an integer, the expression will be an integer

(B) $$\frac{y+z}{x} = \frac{y}{x} + \frac{z}{x}$$ . $$\frac{y}{x}$$ is an integer but we can't say if $$\frac{z}{x}$$ is also an integer. So the expression need not necessarily be an integer.

Although we have got our answer, I am reducing the other expressions for solution purpose.

(C) $$\frac{x+y}{z} = \frac{x}{z} + \frac{y}{z}$$. Both $$\frac{x}{z}$$ and $$\frac{y}{z}$$ are integers, hence the expression will also be an integer

(D) $$\frac{xy}{z}$$. Since $$\frac{x}{z}$$ is an integer, the expression will also be an integer.

(E) $$\frac{yz}{x}$$. Since $$\frac{y}{x}$$ is an integer, the expression will also be an integer.

Hope this helps Regards
Harsh
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Re: If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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1
Answer = (B) $$\frac{(y+z)}{x}$$

Took x = 6; y = 12; z = 2

6 is a factor of 12, and 6 is a multiple of 2

Only option B contradicts the condition
##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 64314
Re: If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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SOLUTION

If x, y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

(A) (x+z)/z
(B) (y+z)/x
(C) (x+y)/z
(D) (xy)/z
(E) (yz)/x

Given: $$z$$ goes into $$x$$ and $$x$$ goes into $$y$$. Note that it's not necessarily means that $$z<x<y$$, it means that $$z\leq{x}\leq{y}$$ (for example all three can be equal x=y=z=1);

Now, in all options but B we can factor out the denominator from the nominator and reduce it. For example in A: $$\frac{x+z}{z}$$ as $$z$$ goes into $$x$$ we can factor out it and reduce to get an integer result (or algebraically as $$x=zk$$ for some positive integer $$k$$ then $$\frac{x+z}{z}=\frac{zk+z}{z}=\frac{z(k+1)}{z}=k+1=integer$$).

But in B. $$\frac{y+z}{x}$$ we can not be sure that we'll be able factor out $$x$$ from $$z$$ thus this option might not be an integer (for example x=y=4 and z=2).

Alternately you could juts plug some smart numbers and the first option which would give a non-integer result would be the correct choice.
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Re: If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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3
IMO B.
let y = 10
then x = 2 or 5.
z can be 1 or 2

B satisfies.
Director  G
Joined: 23 Jan 2013
Posts: 506
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Re: If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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1
1
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x, y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

(A) (x+z)/z
(B) (y+z)/x
(C) (x+y)/z
(D) (xy)/z
(E) (yz)/x

Problem Solving
Question: 172
Category: Arithmetic Properties of numbers
Page: 85
Difficulty: 600

GMAT Club is introducing a new project: The Official Guide For GMAT® Quantitative Review, 2ND Edition - Quantitative Questions Project

Each week we'll be posting several questions from The Official Guide For GMAT® Quantitative Review, 2ND Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

We'll be glad if you participate in development of this project:
2. Please vote for the best solutions by pressing Kudos button;
3. Please vote for the questions themselves by pressing Kudos button;
4. Please share your views on difficulty level of the questions, so that we have most precise evaluation.

Thank you!

We have integers Y,X and Z multiples of Z, all not equal to 0 and Y>=X>=Z

(A) (x+z)/z: we add multiples of Z and divide it by Z, it is always integer
(B) (y+z)/x: it is integer when Y=X=Z and Y>X=Z and not integer when Y>X>Z and Y=X>Z
(C) (x+y)/z: the same as in A
(D) (xy)/z: if one multiple of other, multiplying with any other guarantees multiple
(E) (yz)/x: the same as in D

For test strategy we could use picking but it has danger for testing B. We can choose B by exluding ACDE
Manager  Joined: 20 Dec 2013
Posts: 114
Re: If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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[quote="Bunuel"]The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x, y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

(A) (x+z)/z
(B) (y+z)/x
(C) (x+y)/z
(D) (xy)/z
(E) (yz)/x

If we plug in: x = 6, y = 12 and z = 3.

A - (9)/3
B - (15)/12 - NOT an integer
C - (18)/3
D - Integer
E - Integer

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

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Hi All,

This question is perfect for TESTing VALUES. Since the prompt asks 'which of the following is NOT necessarily an integer?", we can take advantage of a great 'shortcut' - we just need to find ONE instance for any answer to NOT be an integer and we'll have the solution...

We're given a few facts to work with:
1) X, Y and Z are POSITIVE INTEGERS
2) X is a FACTOR of Y
3) X is a MULTIPLE of Z

Let's TEST VALUES.....

IF....
X = 2
Y = 2
Z = 1

Answer A: (X+Z)/Z = (2+1)/1 = 3 This is an integer
Answer B: (Y+Z)/X = (2+1)/2 = 3/2 This is NOT an integer, so this MUST be the answer. We don't even have to check the others.

GMAT assassins aren't born, they're made,
Rich
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Re: If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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1
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x, y, and z are positive integers such that x is a factor of y, and x is a multiple of z, which of the following is NOT necessarily an integer?

(A) (x+z)/z
(B) (y+z)/x
(C) (x+y)/z
(D) (xy)/z
(E) (yz)/x

Problem Solving
Question: 172
Category: Arithmetic Properties of numbers
Page: 85
Difficulty: 600

GMAT Club is introducing a new project: The Official Guide For GMAT® Quantitative Review, 2ND Edition - Quantitative Questions Project

Each week we'll be posting several questions from The Official Guide For GMAT® Quantitative Review, 2ND Edition and then after couple of days we'll provide Official Answer (OA) to them along with a slution.

We'll be glad if you participate in development of this project:
2. Please vote for the best solutions by pressing Kudos button;
3. Please vote for the questions themselves by pressing Kudos button;
4. Please share your views on difficulty level of the questions, so that we have most precise evaluation.

Thank you!

An easy approach would be to bring everything in terms of z.

x = za (where a is a positive integer)
y = xb = zab (where b is a positive integer)

(A) (x+z)/z = (za + z)/z = a + 1 (positive integer)
(B) (y+z)/x = (zab + z)/za = (ab + 1)/a (Not necessarily an integer)
(C) (x+y)/z = (zab + za)/z = ab + a (positive integer)
(D) (xy)/z = zazab/z = zaab (positive integer)
(E) (yz)/x = zabz/za = zb (positive integer)

_________________
Karishma
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Re: If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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Hey everyone,

this question took me 7 minutes to solve correctly, is there any shortcut for dummies to solve this quckly ?  here is my solution its only 30sec read, please read it below you wont get bored Let y be 6
Then x = 2 or x =3
Z =1 or z =2

x is a factor of y -----> 2 is factor of 6
x is a multiple of z -----> 2 is multiple of 1

so lets test them; this is the most exciting part of problem solving process (A) (x+z)/z ( 2+2)/2 or (2+1)1
(B) (y+z)/x (6+1)/2= non integer or (6+2)/2 integer
(C) (x+y)/z 2+6/1 or 2+6/2
(D) (xy)/z 2*6/1 or 3*6/2
(E) (yz)/x 6*1/2 , 6*1/2 6*2/3 etc 
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Location: India
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Re: If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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dave13 wrote:
Hey everyone,

this question took me 7 minutes to solve correctly, is there any shortcut for dummies to solve this quckly ? :? :)

here is my solution :) its only 30sec read, please read it below you wont get bored :)

Let y be 6
Then x = 2 or x =3
Z =1 or z =2

x is a factor of y -----> 2 is factor of 6
x is a multiple of z -----> 2 is multiple of 1

so lets test them; this is the most exciting part of problem solving process :)

(A) (x+z)/z ( 2+2)/2 or (2+1)1
(B) (y+z)/x (6+1)/2= non integer or (6+2)/2 integer
(C) (x+y)/z 2+6/1 or 2+6/2
(D) (xy)/z 2*6/1 or 3*6/2
(E) (yz)/x 6*1/2 , 6*1/2 6*2/3 etc 

Hi dave13

In fact what your method i.e. substitution works best for this question and is the fastest route. try $$z=1$$, $$x=2$$ & $$y=4$$. remember from question stem you should have realized that $$y>x>z$$, hence you should pick the smallest number for $$z$$ for easy calculation and then work upwards.

Algebraic approach -

given $$y=kx$$ (where $$k$$ is a constant) and $$x=qz$$ (where $$q$$ is a constant). In the options I could see that only $$x$$ & $$z$$ are in the denominators so

$$\frac{y}{x}=k=integer$$ and $$\frac{x}{z}=q=integer$$. with this understanding scan the options

A) $$\frac{(x+z)}{z}=\frac{x}{z}+\frac{z}{z}=integer+1=integer$$

(B) $$\frac{(y+z)}{x}=\frac{y}{x}+\frac{z}{x}=integer+non-integer=non-integer$$

(C) $$\frac{(x+y)}{z}=\frac{x}{z}+\frac{y}{z}=integer+integer=integer$$

(D) $$\frac{(xy)}{z}=y*\frac{x}{z}=y*integer=integer$$

(E) $$\frac{(yz)}{x}=\frac{y}{x}*z=integer*z=integer$$

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

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Hi niks18,

You have to be careful about the assumptions you make about the given information. At NO point does the prompt state that Y > X.

In my approach, I used X=2, Y=2, Z=1 and found the correct answer WITHOUT having to check 3 of the options.

GMAT assassins aren't born, they're made,
Rich
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Re: If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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plug in: x = 6, y = 12 and z = 3. you will get answer.
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Re: If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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Consider

1. X = BZ
2. Y = AX ; where A & B are integers.

A. (X+Z)/Z = (BZ+Z)/Z = B+1 (integer)

C . (X+Y)/Z = (BZ+ABZ)/Z = B+AB (integer)

D. XY /Z = BZ* Y /Z = BY (integer)

E. YZ /X = AXZ/X = AZ (integer)

B. (Y +Z)/X = (AX + X /B)/X = (AB+1)/B = NOT necessarily integer.
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Re: If x, y, and z are positive integers such that x is a factor of y, and  [#permalink]

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Since x is a factor of y; let y = ax
Also, x is a multiple of z, let x = bz

(A) (x+z)/z ==> (bz +z)/z ==> z (b+1)/z ==> Definitely an integer
(B) (y+z)/x ==> (ax +x/b)/x ==> x (a+1/b)/x ==> Not an integer
(C) (x+y)/z ==>(x+ax)/z ==> x(1+a)/z => bz(1+a)/z ==> Definitely an integer
(D) (xy)/z ==> (bz*y)/z ==>Definitely an integer
(E) (yz)/x ==> (ax*z)/x => Definitely an integer
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Cheers,
NJ Re: If x, y, and z are positive integers such that x is a factor of y, and   [#permalink] 24 May 2020, 23:58

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