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]

31 Aug 2010, 01:22

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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: Common divisor of x and y [#permalink]

31 Aug 2010, 02:14

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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.

answer:B

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Re: Common divisor of x and y [#permalink]

31 Aug 2010, 04:52

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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.

Answer: B.

Hope it's clear.
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Re: Common divisor of x and y [#permalink]

31 Aug 2010, 13:14

Wow, is that a 700 question?
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Re: Common divisor of x and y [#permalink]

21 Sep 2010, 12:32

kalrac wrote:

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

Thanks!
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Re: Common divisor of x and y [#permalink]

21 Sep 2010, 12:39

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metallicafan wrote:

kalrac wrote:

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: Common divisor of x and y [#permalink]

16 Oct 2010, 08:40

great question...

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Re: Common divisor of x and y [#permalink]

16 Feb 2011, 14:23

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: Common divisor of x and y [#permalink]

16 Feb 2011, 14:34

abmyers wrote:

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]

01 Mar 2013, 04:39

I fail

not easy at all

I want to follow this posting.

Re: If x and y are positive integers such that x = 8y + 12, what [#permalink] 01 Mar 2013, 04:39

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If x and y are positive integers such that x = 8y + 12, what

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