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If x and y are positive integers, what is the greatest common divisor

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zest4mba
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If x and y are positive integers, what is the greatest common divisor [#permalink] New post   23 Aug 2010, 11:27
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If x and y are positive integers, what is the greatest common divisor of x and y?

(1) 2x + y = 73
(2) 5x – 3y = 1
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Bunuel
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   04 Mar 2012, 07:41
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carcass wrote:
If x and y are positive integers, what is the greatest common divisor of x and y?

1) 2x + y = 73
2) 5x – 3y = 1

MMMMMMmm

Here I'm not sure that the answer is C because is true that we need of both statement to find possible values for X and Y. Infact statement 1 and 2 we do not have values for the variables (can be everything).

But it seems to be a trap answer......


If x and y are positive integers, what is the greatest common divisor of x and y?

This is a classic "C trap" question: "C trap" is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together. When you see such question you should be extremely cautious when choosing C for an answer.

(1) \(2x+y=73\). Suppose GCD(x, y) is some integer \(d\), then \(x=md\) and \(y=nd\), for some positive integers \(m\) and \(n\). So, we'll have \(2(md)+(nd)=d(2m+n)=73\). Now, since 73 is a prime number (73=1*73) then \(d=1\) and \(2m+n=73\) (vice versa is not possible because \(m\) and \(n\) are positve integers and therefore \(2m+n\) cannot equal to 1). Hence we have that GCD(x, y)=d=1. Sufficient.

(2) \(5x-3y=1\) --> \(5x=3y+1\) --> \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.

Answer: D.

Hope it's clear.
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   23 Aug 2010, 13:00
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zest4mba wrote:
If x and y are positive integers, what is the greatest common divisor of x and y?

1. 2x + y = 73
2. 5x – 3y = 1


This is a classic "C trap" question: "C trap" is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together. When you see such question you should be extremely cautious when choosing C for an answer.

(1) \(2x+y=73\). Suppose GCD(x, y) is some integer \(d\), then \(x=md\) and \(y=nd\), for some positive integers \(m\) and \(n\). So, we'll have \(2(md)+(nd)=d(2m+n)=73\). Now, since 73 is a prime number (73=1*73) then \(d=1\) and \(2m+n=73\) (vice versa is not possible since \(m\) and \(n\) are positve integers and therefore \(2m+n\) cannot equal to 1). Hence we have that GCD(x, y)=d=1. Sufficient.

(2) \(5x-3y=1\) --> \(5x=3y+1\). So \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1. So \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.

Answer: D.

Hope it's clear.
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   23 Aug 2010, 12:58
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zest4mba wrote:
If x and y are positive integers, what is the greatest common divisor of x and y?

1. 2x + y = 73
2. 5x – 3y = 1


Say x and y were both divisible by some number d. Then 2x + y would certainly be a multiple of d (if you add two multiples of d, you always get a multiple of d). Now we know from statement 1 that 2x + y is the number 73, so if 2x+y is divisible by d, then 73 must be divisible by d. But 73 is prime, so d could only be 1 or 73. Clearly d can't be 73, since then 2x +y would not equal 73, so the only possible value of d is 1, and thus 1 is the only common divisor of x and y.

You can use the same logic for statement 2: If x and y are both multiples of d, then 5x - 3y would need to be a multiple of d. But 5x-3y = 1, so 1 is a multiple of d, and d must be 1.

D.
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   04 Mar 2012, 09:22
Thanks bunuel.

From your explanation can we suppose to apply the reasoning from statement two to the first one or is not possible ?? would be a big mistake because 1 is NOT prime??'
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   05 Mar 2012, 02:28
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carcass wrote:
Thanks bunuel.

From your explanation can we suppose to apply the reasoning from statement two to the first one or is not possible ?? would be a big mistake because 1 is NOT prime??'


I suppose you mean whether we can apply the reasoning from (1) to statement (2). Yes, we can:

(2) \(5x-3y=1\) --> Suppose GCD(x, y) is some integer \(d\), then \(x=md\) and \(y=nd\), for some positive integers \(m\) and \(n\). So, we'll have \(5(md)-3(nd)=d(5m-3n)=1\) --> \(d\) is a factor of 1, so \(d\) must equal 1. Sufficient.
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   05 Sep 2012, 08:24
Bunuel wrote:
carcass wrote:
If x and y are positive integers, what is the greatest common divisor of x and y?

1) 2x + y = 73
2) 5x – 3y = 1

MMMMMMmm

Here I'm not sure that the answer is C because is true that we need of both statement to find possible values for X and Y. Infact statement 1 and 2 we do not have values for the variables (can be everything).

But it seems to be a trap answer......


If x and y are positive integers, what is the greatest common divisor of x and y?

This is a classic "C trap" question: "C trap" is a problem which is VERY OBVIOUSLY sufficient if both statements are taken together. When you see such question you should be extremely cautious when choosing C for an answer.

(1) \(2x+y=73\). Suppose GCD(x, y) is some integer \(d\), then \(x=md\) and \(y=nd\), for some positive integers \(m\) and \(n\). So, we'll have \(2(md)+(nd)=d(2m+n)=73\). Now, since 73 is a prime number (73=1*73) then \(d=1\) and \(2m+n=73\) (vice versa is not possible because \(m\) and \(n\) are positve integers and therefore \(2m+n\) can not equal to 1). Hence we have that GCD(x, y)=d=1. Sufficient.

(2) \(5x-3y=1\) --> \(5x=3y+1\) --> \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.

Answer: D.

Hope it's clear.



Hi,

Thanks for the explanation.

I m not so good in reasoning so I put the values and check.

For example in statement 1 I just entered the values of y as 1,3,5,7 and I get the values of X as 36,35,34,33 respect. After checking 4 - 5 values I got to know that the gcf is 1.


Please let me know if my strategy is good or not.

Thanks,
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   05 Sep 2012, 22:05
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vivekdixit07 wrote:
Hi,

Thanks for the explanation.

I m not so good in reasoning so I put the values and check.

For example in statement 1 I just entered the values of y as 1,3,5,7 and I get the values of X as 36,35,34,33 respect. After checking 4 - 5 values I got to know that the gcf is 1.


Please let me know if my strategy is good or not.

Thanks,


Try to understand this: If something generic is established, you can plug in specific examples and use them e.g. if I say, "All boys are crazy." I can say, "Tom, a boy, is crazy."
But the other way around may not always work. From certain examples, you cannot establish something generic. e.g. I cannot say, "Tom is crazy. Alfred is crazy. Ross is crazy. We can conclude that all boys are crazy."
Here, you have tried to do something like this which is not good. We cannot blindly plug in numbers and establish that GCD will be 1. We could have missed some pairs where GCD may not have been 1. What you can do is plug in some numbers and then think why you are getting GCD = 1 in each case and whether it will be true for all pair of values.

Check out Bunuel's explanation above. It's very important and you can expect to be tested on such concepts in GMAT.
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   15 Sep 2012, 17:38
VeritasPrepKarishma wrote:
vivekdixit07 wrote:
Hi,

Thanks for the explanation.

I m not so good in reasoning so I put the values and check.

For example in statement 1 I just entered the values of y as 1,3,5,7 and I get the values of X as 36,35,34,33 respect. After checking 4 - 5 values I got to know that the gcf is 1.


Please let me know if my strategy is good or not.

Thanks,


Try to understand this: If something generic is established, you can plug in specific examples and use them e.g. if I say, "All boys are crazy." I can say, "Tom, a boy, is crazy."
But the other way around may not always work. From certain examples, you cannot establish something generic. e.g. I cannot say, "Tom is crazy. Alfred is crazy. Ross is crazy. We can conclude that all boys are crazy."
Here, you have tried to do something like this which is not good. We cannot blindly plug in numbers and establish that GCD will be 1. We could have missed some pairs where GCD may not have been 1. What you can do is plug in some numbers and then think why you are getting GCD = 1 in each case and whether it will be true for all pair of values.

Check out Bunuel's explanation above. It's very important and you can expect to be tested on such concepts in GMAT.



===============
Karishma, Bunuel,

I solved the above problem in the following manner. Please let me know whether that is the right approach or not.

1) 2x+y=73=>y=73-2x
As x & y are positive integers=> let x= 1 => y=71
x=2=> y=69
x=3=> y=67
=> x & y have no common factors other than 1 in each cases. Sufficient
2) 5x-3y=1=> y=(5x-1)/3
As x& y are +ve integers for x= 2, y=3
x=5, y=8
x=8, y=13
=> In each of the above cases, x & y have GCF=1. Sufficient

ANS- D
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   16 Sep 2012, 23:58
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monsoon1 wrote:

I solved the above problem in the following manner. Please let me know whether that is the right approach or not.

1) 2x+y=73=>y=73-2x
As x & y are positive integers=> let x= 1 => y=71
x=2=> y=69
x=3=> y=67
=> x & y have no common factors other than 1 in each cases. Sufficient
2) 5x-3y=1=> y=(5x-1)/3
As x& y are +ve integers for x= 2, y=3
x=5, y=8
x=8, y=13
=> In each of the above cases, x & y have GCF=1. Sufficient

ANS- D


Ok, let me give you an example to show you that this approach alone is not good.

Say, I change the first statement a little:

1. 2x + y = 35
As x & y are positive integers=> let x= 1 => y=33
x=2=> y=31
x=3=> y=29

x &y have GCF = 1 in each case. Does it mean they will have GCF = 1 only?
What if x = 10, y = 15? These values satisfy 2x + y = 35 but the GCF is not 1.

Point is, after how many values do you say that for all values GCF will be 1?

Plugging in numbers can help you think straight but it may not give you the correct answer. That is why understanding the theory is important.
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   18 May 2014, 00:59
Bunuel wrote:
carcass wrote:
If x and y are positive integers, what is the greatest common divisor of x and y?


(2) \(5x-3y=1\) --> \(5x=3y+1\) --> \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1.

Answer: D.

Hope it's clear.



Hi Bunnel,

Could you please explain how \(5x\) and \(3y\) are consecutive integers? I'm confused on this part.

Thanks!
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   18 May 2014, 01:29
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pretzel wrote:
Bunuel wrote:
carcass wrote:
If x and y are positive integers, what is the greatest common divisor of x and y?


(2) \(5x-3y=1\) --> \(5x=3y+1\) --> \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1.

Answer: D.

Hope it's clear.



Hi Bunnel,

Could you please explain how \(5x\) and \(3y\) are consecutive integers? I'm confused on this part.

Thanks!


\(5x=3y+1\) means that 5x is 1 more than 3y, thus \(5x\) and \(3y\) are consecutive integers.

Does this make sense?
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   06 Jul 2014, 23:32
Maybe a silly question but how do we realize that these questions are trap questions and not simple questions. I must admit that though I got Bunuel's approach once I saw it , I will have never have been able to put that at first. During actual GMAT , I wonder if anyone will use this approach. Are there any obvious giveaways which could give a hint about thinking on different lines ? Do we have to see at what point in the exam is the question coming to be able to figure out whether it is a trap one or simple one ?
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   07 Jul 2014, 01:12
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himanshujovi wrote:
Maybe a silly question but how do we realize that these questions are trap questions and not simple questions. I must admit that though I got Bunuel's approach once I saw it , I will have never have been able to put that at first. During actual GMAT , I wonder if anyone will use this approach. Are there any obvious giveaways which could give a hint about thinking on different lines ? Do we have to see at what point in the exam is the question coming to be able to figure out whether it is a trap one or simple one ?


As I wrote both statements taken together are VERY OBVIOUSLY sufficient to answer the question. When you see such question you should be extremely cautious when choosing C for an answer.
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   07 Jul 2014, 02:47
Exactly Bunuel.. That was a giveaway but again that would be contextual, right. For eg in a CAT like GMAT, if this question comes as the first question, cant one be presume it to be the simplest question which GMAC is using to slowly build up the adaptive level for the test taker. Obviously if it comes at the very end and one is going great guns then the concept of CAT will kick in and one will try and figure out the trick in it


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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   07 Jul 2014, 06:02
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himanshujovi wrote:
Exactly Bunuel.. That was a giveaway but again that would be contextual, right. For eg in a CAT like GMAT, if this question comes as the first question, cant one be presume it to be the simplest question which GMAC is using to slowly build up the adaptive level for the test taker. Obviously if it comes at the very end and one is going great guns then the concept of CAT will kick in and one will try and figure out the trick in it


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I have known people to get stuck on their first or second question so I wouldn't go to the test with any presumptions. But yeah, I don't think it will be the very first question. But after 3-4 questions, all questions are fair play.
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   08 Jul 2014, 01:37
If 5x and 3y are consecutive, does that mean x and y will always b consecutive too?
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   08 Jul 2014, 02:02
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manish2014 wrote:
If 5x and 3y are consecutive, does that mean x and y will always b consecutive too?


No. x and y will be consecutive if x=2 and y=3 or x=-1 and y=-2. But in all other cases x and y won't be consecutive, for example, x=5 and y=8.
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   18 Aug 2014, 11:57
Bunuel wrote:
\(5x-3y=1\) --> \(5x=3y+1\) --> \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.


Hope it's clear.

Hi Bunuel,
Although 5x and 3y are consecutive integers and co-prime , why are x and y co-prime?
is it because 5 and 3 are also co-prime?
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Re: If x and y are positive integers, what is the greatest common divisor [#permalink] New post   18 Aug 2014, 12:21
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ronr34 wrote:
Bunuel wrote:
\(5x-3y=1\) --> \(5x=3y+1\) --> \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1 (for example 20 and 21 are consecutive integers, thus only common factor they share is 1). So, \(5x\) and \(3y\) don't share any common factor but 1, thus \(x\) and \(y\) also don't share any common factor but 1. Hence, GCD(x, y) is 1. Sufficient.


Hope it's clear.

Hi Bunuel,
Although 5x and 3y are consecutive integers and co-prime , why are x and y co-prime?
is it because 5 and 3 are also co-prime?


Let me ask you a question: if x and y shared any common factor but 1, would 5x and 3y be co-prime? Wouldn't they also share that factor?
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