If x and y are positive integers such that x = 8y + 12, what : GMAT Data Sufficiency (DS)
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# If x and y are positive integers such that x = 8y + 12, what

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If x and y are positive integers such that x = 8y + 12, what [#permalink]

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31 Aug 2010, 00:22
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If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?

(1) x = 12u, where u is an integer
(2) y = 12z, where z is an integer
[Reveal] Spoiler: OA

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Re: If x and y are positive integers such that x = 8y + 12, what [#permalink]

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31 Aug 2010, 01:14
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Stat 1: x= 12u returns
x= 12u and y=3/2(u-1)
GCD of x and y varies for u=0 and u is +ve

Stat 2: y=12z returns
x=12(8z+1),y=12z
GCD for any inter value of z is 12.

Hence statement 2 alone is sufficient.

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Re: If x and y are positive integers such that x = 8y + 12, what [#permalink]

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31 Aug 2010, 03:52
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metallicafan wrote:
If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?
(1) x = 12u, where u is an integer
(2) y = 12z, where z is an integer

Given: $$x=8y+12$$.

(1) $$x=12u$$ --> $$12u=8y+12$$ --> $$3(u-1)=2y$$ --> the only thing we know from this is that 3 is a factor of $$y$$. Is it GCD of $$x$$ and $$y$$? Not clear: if $$x=36$$, then $$y=3$$ and $$GCD(x,y)=3$$ but if $$x=60$$, then $$y=6$$ and $$GCD(x,y)=6$$ --> two different answers. Not sufficient.

(2) $$y=12z$$ --> $$x=8*12z+12$$ --> $$x=12(8z+1)$$ --> so 12 is a factor both $$x$$ and $$y$$.

Is it GCD of $$x$$ and $$y$$? Why can not it be more than 12, for example 13, 16, 24, ... We see that factors of $$x$$ are 12 and $$8z+1$$: so if $$8z+1$$ has some factor >1 common with $$z$$ then GCD of $$x$$ and $$y$$ will be more than 12 (for example if $$z$$ and $$8z+1$$ are multiples of 5 then $$x$$ would be multiple of $$12*5=60$$ and $$y$$ also would be multiple of $$12*5=60$$, so GCD of $$x$$ and $$y$$ would be more than 12). But $$z$$ and $$8z+1$$ CAN NOT share any common factor >1, as $$8z+1$$ is a multiple of $$z$$ plus 1, so no factor of $$z$$ will divide $$8z+1$$ evenly, which means that GCD of $$x$$ and $$y$$ can not be more than 12. $$GCD(x,y)=12$$. Sufficient.

Hope it's clear.
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Re: If x and y are positive integers such that x = 8y + 12, what [#permalink]

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31 Aug 2010, 12:14
Wow, is that a 700 question?
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Re: If x and y are positive integers such that x = 8y + 12, what [#permalink]

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21 Sep 2010, 11:32
kalrac wrote:
Stat 1: x= 12u returns
x= 12u and y=3/2(u-1)
GCD of x and y varies for u=0 and u is +ve

Stat 2: y=12z returns
x=12(8z+1),y=12z
GCD for any inter value of z is 12.

Hence statement 2 alone is sufficient.

I don't understand the quoted solution, can someone please explain it?

Thanks!
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Re: If x and y are positive integers such that x = 8y + 12, what [#permalink]

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21 Sep 2010, 11:39
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metallicafan wrote:
kalrac wrote:
Stat 1: x= 12u returns
x= 12u and y=3/2(u-1)
GCD of x and y varies for u=0 and u is +ve

Stat 2: y=12z returns
x=12(8z+1),y=12z
GCD for any inter value of z is 12.

Hence statement 2 alone is sufficient.

I don't understand the quoted solution, can someone please explain it?

Thanks!

I think the quoted solution refers to the following rule: if $$a$$ and $$b$$ are multiples of $$k$$ and are $$k$$ units apart from each other then $$k$$ is greatest common divisor of $$a$$ and $$b$$.

For example if $$a$$ and $$b$$ are multiples of 7 and $$a=b+7$$ then 7 is GCD of $$a$$ and $$b$$.

So if we apply this rule to (2) we would have: both $$x$$ and $$y$$ are multiple of 12 and are 12 apart each other, so 12 is GCD of $$x$$ and $$y$$.
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Re: If x and y are positive integers such that x = 8y + 12, what [#permalink]

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16 Oct 2010, 07:40
great question...
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Re: If x and y are positive integers such that x = 8y + 12, what [#permalink]

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16 Feb 2011, 13:23
Bunuel,

You wrote:

(1) x=12u --> 12u=8y+12 --> 3(u-1)=2y --> the only thing we know from this is that 3 is a multiple of y.

How do we know that 3 is a muliple of y? I mean, I worked it out by plugging in values for u and found that it is true, but is there some property of the formula that gives it away?
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Re: If x and y are positive integers such that x = 8y + 12, what [#permalink]

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16 Feb 2011, 13:34
abmyers wrote:
Bunuel,

You wrote:

(1) x=12u --> 12u=8y+12 --> 3(u-1)=2y --> the only thing we know from this is that 3 is a multiple of y.

How do we know that 3 is a muliple of y? I mean, I worked it out by plugging in values for u and found that it is true, but is there some property of the formula that gives it away?

3(u-1)=2y --> the only thing we know from this is that 3 is a factor of y --> 3(u-1) is a multiple of 3, so must be 2y as they are equal. Now, 2y to be multiple of 3 then y must be multiple of 3.
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Re: If x and y are positive integers such that x = 8y + 12, what [#permalink]

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01 Mar 2013, 03:39
I fail

not easy at all

I want to follow this posting.
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Re: If x and y are positive integers such that x = 8y + 12, what [#permalink]

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15 Mar 2014, 18:15
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Re: If x and y are positive integers such that x = 8y + 12, what [#permalink]

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25 Mar 2014, 09:38
x = 8y + 12 simplifies to x=4(2y+3)

Statement 1:
substituting x=12u

12u=4(2y+3)
3u=2y-3
3(u-1)=2y

So u should be greater than 1 and should be odd (because u-1 is a multiple of 2)
Checking for values

u x y GCD
3 36 3 3
5 60 6 6

So statement 1 not sufficient

Statement 2:
substituting y=12z

x=4(12z+3)
x=12(4z+1)

Checking for values

z x y GCD
1 60 12 12
2 108 24 12
3 154 36 12
we can check for more values but usually we would see any variations withing 1st 3-4 values.

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Re: If x and y are positive integers such that x = 8y + 12, what [#permalink]

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05 Oct 2015, 16:15
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Re: If x and y are positive integers such that x = 8y + 12, what   [#permalink] 05 Oct 2015, 16:15
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