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

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If x and y are positive integers, what is the greatest [#permalink] New post 04 Mar 2012, 06:21
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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

[Reveal] Spoiler:
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......
[Reveal] Spoiler: OA

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Last edited by Bunuel on 09 Jul 2014, 09:50, edited 2 times in total.
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Re: If x and y are positive integers, what is the greatest [#permalink] New post 04 Mar 2012, 06:41
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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 [#permalink] New post 04 Mar 2012, 08:22
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Re: If x and y are positive integers, what is the greatest [#permalink] New post 05 Mar 2012, 01:28
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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 [#permalink] New post 07 Apr 2012, 14:55
Hi Bunuel, Can I hire your mind for my exam...? Well if I get a good grade, a large part of it would be due to you....Thanks.

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.
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Re: If x and y are positive integers, what is the greatest [#permalink] New post 05 Sep 2012, 07:24
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 [#permalink] New post 05 Sep 2012, 13:04
Awesome explanation Brunnel.

Vivek, i tried to solve by plugging in numbers.
for (1) i plugged in 20,33 10,53 15, 43 and so on which satisfy the equation
and noticed that none of the pairs have any common prime factors.
So the GCF has to be 1.
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Re: If x and y are positive integers, what is the greatest [#permalink] New post 05 Sep 2012, 21:05
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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 [#permalink] New post 15 Sep 2012, 16:38
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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.



===============
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 [#permalink] New post 16 Sep 2012, 22:58
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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 [#permalink] New post 09 Jul 2013, 14:38
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Re: If x and y are positive integers, what is the greatest [#permalink] New post 17 May 2014, 23:59
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 [#permalink] New post 18 May 2014, 00:29
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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 [#permalink] New post 06 Jul 2014, 22:32
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 [#permalink] New post 07 Jul 2014, 00:12
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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 [#permalink] New post 07 Jul 2014, 01:47
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 [#permalink] New post 07 Jul 2014, 05:02
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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 [#permalink] New post 08 Jul 2014, 00:37
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 [#permalink] New post 08 Jul 2014, 01:02
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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 [#permalink] New post 08 Jul 2014, 06:27
out of all the questions which I have encountered in my GMAT prep , this is the scariest.. Mind you not the toughest but still the scariest
Re: If x and y are positive integers, what is the greatest   [#permalink] 08 Jul 2014, 06:27
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