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# If x and y are positive integers, which of the following

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Manager
Joined: 30 Dec 2008
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If x and y are positive integers, which of the following  [#permalink]

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Updated on: 25 May 2013, 03:49
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If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y?

A. 5
B. 5(x-y)
C. 20x
D. 20y
E. 35x

Originally posted by cul3s on 18 Jan 2009, 21:40.
Last edited by Bunuel on 25 May 2013, 03:49, edited 2 times in total.
Edited the question and added the OA
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Posts: 50615

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08 Feb 2012, 02:06
19
21
If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y?
A. 5
B. 5(x – y)
C. 20x
D. 20y
E. 35x

Greatest common divisor (GCD) of $$35x$$ and $$20y$$ obviously must be a divisor of both $$35x$$ and $$20y$$, which means that $$\frac{35x}{GCD}$$ and $$\frac{20y}{GCD}$$ must be an integer.

If $$GCD=20x$$ (option C), then $$\frac{35x}{20x}=\frac{7}{4}\neq{integer}$$, which means that $$20x$$ cannot be GCD of $$35x$$ and $$20y$$ as it is not a divisor of $$35x$$.

How about the other choices, can they be GCD of $$35x$$ and $$20y$$?

A. $$5$$ --> if $$x=y=1$$ --> $$35x=35$$ and $$20y=20$$ --> $$GCD(35,20)=5$$. Answer is YES, $$5$$ can be GCD of $$35x=35$$ and $$20y$$;

B. $$5(x-y)$$ --> if $$x=3$$ and $$y=2$$ --> $$35x=105$$ and $$20y=40$$ --> $$GCD(105,40)=5=5(x-y)$$. Answer is YES, $$5(x-y)$$ can be GCD of $$35x$$ and $$20y$$;

D. $$20y$$ --> if $$x=4$$ and $$y=1$$ --> $$35x=140$$ and $$20y=20$$ --> $$GCD(140,20)=20=20y$$. Answer is YES, $$20y$$ can be GCD of $$35x$$ and $$20y$$;

E. $$35x$$ --> if $$x=1$$ and $$y=7$$ --> $$35x=35$$ and $$20y=140$$ --> $$GCD(35,140)=35=35x$$. Answer is YES, $$35x$$ can be GCD of $$35x$$ and $$20y$$.

Hope it's clear.
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19 Jan 2009, 08:38
7
3
If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y?

a. 5
b. 5(x-y)
c. 20x
d. 20y
e. 35x

We are looking for a choice that CANNOT be the greatest common divisor of 35x and 20y ...which means 35x and 20y when divided by the answer choice the quotient should not be a integer.
lets check

a. 5 35x/5 = 7x and 20y/5 = 4y both are integers so eliminate
b. 5(x-y) when x = 2 and y = 1 it could be be the greatest common divisor ..so eliminate
c. 20x when x = 1 its 20 and 20 cannot be the greatest common divisor of 35x and 20y ...
or 35x/20x = 7/4 which is not a integer.

##### General Discussion
Director
Joined: 29 Nov 2012
Posts: 765
Re: If x and y are positive integers, which of the following  [#permalink]

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12 Jul 2013, 08:15
would plugging in number a better strategy for such problems?
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Re: If x and y are positive integers, which of the following  [#permalink]

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12 Jul 2013, 08:16
2
fozzzy wrote:
would plugging in number a better strategy for such problems?

Better strategy is the one that suits you best.
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12 Jul 2013, 08:26
Bunuel wrote:
If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y?
A. 5
B. 5(x – y)
C. 20x
D. 20y
E. 35x

Greatest common divisor (GCD) of $$35x$$ and $$20y$$ obviously must be a divisor of both $$35x$$ and $$20y$$, which means that $$\frac{35x}{GCD}$$ and $$\frac{20y}{GCD}$$ must be an integer.

If $$GCD=20x$$ (option C), then $$\frac{35x}{20x}=\frac{7}{4}\neq{integer}$$, which means that $$20x$$ cannot be GCD of $$35x$$ and $$20y$$ as it is not a divisor of $$35x$$.

In this question for the division does it mean that both X and Y must be divisible or if any one is divisible the solution works

In option D

$$\frac{35X}{20Y}$$ doesn't work according to that strategy?
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Re: If x and y are positive integers, which of the following  [#permalink]

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12 Jul 2013, 08:32
1
I proceeded like this:

35x can have following prime factors : 5 ,7, x [well, x can have > 1 prime factors too; if x=6, 2 and 3 will be added to the list of prime factors]

Similarly, 20y has following prime factors : 2, 5, y [Same theory holds good for y]

the GCF has to have one 5 for sure. [IF we found any answer choices that is not a multiple of 5, it could be omitted right away]

A. 5 => We already covered that GCF has 5. Eliminate
B. 5 (x -y) => If x and y were 2 and 1 respectively, this would reduce to 5. (same as answer choice A). Eliminate.
C. 20x prime factors are 2, 5 and x. For 2 to be part of GCF, it must have come from x as 35 in 35x doesn't have 2.
[If x had 2's then, 20x= 4 x 5 x X as GCF would not tally because, there is only two 2's in 20y]
D. 20y = 2 x 2 x 5 x y ... If x were 4, this would be very possible.
E. 35x = 5 * 7 * x; If y=7 and x =4, this is also possible.

There are simpler reasons already stated to say why C is the answer. But, for those who use prime factor trees to attach such problems, this is how I would explain.
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Updated on: 12 Jul 2013, 08:39
2
fozzzy wrote:
Bunuel wrote:
If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y?
A. 5
B. 5(x – y)
C. 20x
D. 20y
E. 35x

Greatest common divisor (GCD) of $$35x$$ and $$20y$$ obviously must be a divisor of both $$35x$$ and $$20y$$, which means that $$\frac{35x}{GCD}$$ and $$\frac{20y}{GCD}$$ must be an integer.

If $$GCD=20x$$ (option C), then $$\frac{35x}{20x}=\frac{7}{4}\neq{integer}$$, which means that $$20x$$ cannot be GCD of $$35x$$ and $$20y$$ as it is not a divisor of $$35x$$.

In this question for the division does it mean that both X and Y must be divisible or if any one is divisible the solution works

In option D

$$\frac{35X}{20Y}$$ doesn't work according to that strategy?

hi fozzy ,

i will say that best way to undersatand the defenetions of GCF and LCM.

GCF of 2 numbers means ...biggest number which is factor of those numbers.

now hers 35x==>prime factors 5/7...and others we dont know about x
now 20y==>prime factors 2/2/5..and others we dont know as we dont about y

now as lets take options C:
LETS SAY 20x is GCF...THEN IT MUST BE FACTOR OF BOTH...means..==>35x/20x==>this must be integer(according to defenetion of factor)==>but when we simplify that we are getting 7/4==>fraction===>hence we are sure 100 percent that this cant be a factor of both....hence it cant be GCF.

in rest all option we unknown variables are not getting cancelled...so we are not sure in that.

hope it helps
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Originally posted by blueseas on 12 Jul 2013, 08:34.
Last edited by blueseas on 12 Jul 2013, 08:39, edited 1 time in total.
Math Expert
Joined: 02 Sep 2009
Posts: 50615

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12 Jul 2013, 08:37
fozzzy wrote:
Bunuel wrote:
If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y?
A. 5
B. 5(x – y)
C. 20x
D. 20y
E. 35x

Greatest common divisor (GCD) of $$35x$$ and $$20y$$ obviously must be a divisor of both $$35x$$ and $$20y$$, which means that $$\frac{35x}{GCD}$$ and $$\frac{20y}{GCD}$$ must be an integer.

If $$GCD=20x$$ (option C), then $$\frac{35x}{20x}=\frac{7}{4}\neq{integer}$$, which means that $$20x$$ cannot be GCD of $$35x$$ and $$20y$$ as it is not a divisor of $$35x$$.

In this question for the division does it mean that both X and Y must be divisible or if any one is divisible the solution works

In option D

$$\frac{35X}{20Y}$$ doesn't work according to that strategy?

Not sure I understand your question...

But notice that $$\frac{35x}{20y}=\frac{7x}{4y}$$ could be an integer, for example if x=4 and y=1.
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23 Nov 2013, 01:37
Bunuel wrote:
If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y?
A. 5
B. 5(x – y)
C. 20x
D. 20y
E. 35x

Greatest common divisor (GCD) of $$35x$$ and $$20y$$ obviously must be a divisor of both $$35x$$ and $$20y$$, which means that $$\frac{35x}{GCD}$$ and $$\frac{20y}{GCD}$$ must be an integer.

If $$GCD=20x$$ (option C), then $$\frac{35x}{20x}=\frac{7}{4}\neq{integer}$$, which means that $$20x$$ cannot be GCD of $$35x$$ and $$20y$$ as it is not a divisor of $$35x$$.

How about the other choices, can they be GCD of $$35x$$ and $$20y$$?

A. $$5$$ --> if $$x=y=1$$ --> $$35x=35$$ and $$20y=20$$ --> $$GCD(35,20)=5$$. Answer is YES, $$5$$ can be GCD of $$35x=35$$ and $$20y$$;

B. $$5(x-y)$$ --> if $$x=3$$ and $$y=2$$ --> $$35x=105$$ and $$20y=40$$ --> $$GCD(105,40)=5=5(x-y)$$. Answer is YES, $$5(x-y)$$ can be GCD of $$35x$$ and $$20y$$;

D. $$20y$$ --> if $$x=4$$ and $$y=1$$ --> $$35x=140$$ and $$20y=20$$ --> $$GCD(140,20)=20=20y$$. Answer is YES, $$20y$$ can be GCD of $$35x$$ and $$20y$$;

E. $$35x$$ --> if $$x=1$$ and $$y=7$$ --> $$35x=35$$ and $$20y=140$$ --> $$GCD(35,140)=35=35x$$. Answer is YES, $$35x$$ can be GCD of $$35x$$ and $$20y$$.

Hope it's clear.

Hi Bunuel,
Is there a way to do this using prime factorization of 35 and 20?
That's the first thing that comes to mind, but I can see how to proceed from there.
Thanks,
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Joined: 15 Aug 2013
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26 May 2014, 11:46
Bunuel wrote:
If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y?
A. 5
B. 5(x – y)
C. 20x
D. 20y
E. 35x

Greatest common divisor (GCD) of $$35x$$ and $$20y$$ obviously must be a divisor of both $$35x$$ and $$20y$$, which means that $$\frac{35x}{GCD}$$ and $$\frac{20y}{GCD}$$ must be an integer.

If $$GCD=20x$$ (option C), then $$\frac{35x}{20x}=\frac{7}{4}\neq{integer}$$, which means that $$20x$$ cannot be GCD of $$35x$$ and $$20y$$ as it is not a divisor of $$35x$$.

How about the other choices, can they be GCD of $$35x$$ and $$20y$$?

A. $$5$$ --> if $$x=y=1$$ --> $$35x=35$$ and $$20y=20$$ --> $$GCD(35,20)=5$$. Answer is YES, $$5$$ can be GCD of $$35x=35$$ and $$20y$$;

B. $$5(x-y)$$ --> if $$x=3$$ and $$y=2$$ --> $$35x=105$$ and $$20y=40$$ --> $$GCD(105,40)=5=5(x-y)$$. Answer is YES, $$5(x-y)$$ can be GCD of $$35x$$ and $$20y$$;

D. $$20y$$ --> if $$x=4$$ and $$y=1$$ --> $$35x=140$$ and $$20y=20$$ --> $$GCD(140,20)=20=20y$$. Answer is YES, $$20y$$ can be GCD of $$35x$$ and $$20y$$;

E. $$35x$$ --> if $$x=1$$ and $$y=7$$ --> $$35x=35$$ and $$20y=140$$ --> $$GCD(35,140)=35=35x$$. Answer is YES, $$35x$$ can be GCD of $$35x$$ and $$20y$$.

Hope it's clear.

Hi Bunuel,

The steps here are easy to follow but one thing that bugs me is the number selection. It's almost as if you had to KNOW the answer to select the numbers to prove the statements worth. On the GMAT, that might be a little challenging.

is there a way to do this algebraically by using Prime Boxes? Meaning, 35 has 7 and 5 as it's PF and 20 has xxx?
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Re: If x and y are positive integers, which of the following  [#permalink]

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25 Jun 2014, 23:14
How i did this (using prime factors/prime boxes)

35x will have following prime factors (pf) : 5 ,7, x (x could be anything but we leave that for now)

20y will have following prime factors (pf) : 2, 5, y (Again y could be anything but we leave that for now)

So : GCF - 5 or 5xy

A. 5 => Eliminate as GCF can be 5

B. 5 (x -y) => Leave the option for now or pick numbers to check. I left it for later (there was no need to come back to this and check as i got C as an answer)

C. 20x = 2 * 2 * 5 * x. GCF could be 5xy but 20y already has two 2's so ideally this should have come from 35x for 2*2 to be in the GCF and hence this is the answer as this can never be the GCF

D. 20y = 2 * 2 * 5 * y ; GCF could be 5xy and if x=4 (we pick this number to prove this option incorrect), this would be true

E. 35x = 5 * 7 * x; GCF could be 5xy and if y=7 (we pick this number to prove this option incorrect), this would be true
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Re: If x and y are positive integers, which of the following  [#permalink]

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14 Mar 2016, 01:24
i was able to arrive at C as for all the value i was able to portray the GCD
but C was not coming to be true
hence I choose C
then i realized then 20x caanot be the GCD as x must be greater than 4x which is impossible.
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Re: If x and y are positive integers, which of the following  [#permalink]

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15 Feb 2018, 12:21
Hi All,

This question can be solved with math "theory" or by TESTing VALUES. Here's how to eliminate the 4 wrong answers by TESTing VALUES...

We're told that X and Y are POSITIVE INTEGERS. We're asked which of the following CANNOT be the greatest common divisor of 35x and 20y.

IF...X = 1, Y = 1...
35 and 20 have a GCD of 5.

IF...X = 3, Y = 2...
105 and 40 have a GCD of 5. 5(3-2) = 5

IF...X = 4, Y = 1...
140 and 20 have a GCD of 20. 20(1) = 2

IF...X = 2, Y = 7...
70 and 140 have a GCD of 70. 35(2) = 70

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Re: If x and y are positive integers, which of the following  [#permalink]

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20 Oct 2018, 22:53
cul3s wrote:
If x and y are positive integers, which of the following CANNOT be the greatest common divisor of 35x and 20y?

A. 5
B. 5(x-y)
C. 20x
D. 20y
E. 35x

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Re: If x and y are positive integers, which of the following &nbs [#permalink] 20 Oct 2018, 22:53
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