Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 09 Feb 2010
Posts: 72

If x and y are positive integers, what is the greatest [#permalink]
Show Tags
23 Aug 2010, 11:27
1
This post received KUDOS
11
This post was BOOKMARKED
Question Stats:
41% (02:59) correct
59% (01:43) wrong based on 1048 sessions
HideShow timer Statistics
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
Official Answer and Stats are available only to registered users. Register/ Login.



GMAT Tutor
Joined: 24 Jun 2008
Posts: 1179

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
23 Aug 2010, 12:58
3
This post received KUDOS
Expert's post
1
This post was BOOKMARKED
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 5x3y = 1, so 1 is a multiple of d, and d must be 1. D.
_________________
GMAT Tutor in Toronto
If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com



Math Expert
Joined: 02 Sep 2009
Posts: 39662

If x and y are positive integers, what is the greatest [#permalink]
Show Tags
23 Aug 2010, 13:00
4
This post received KUDOS
Expert's post
12
This post was BOOKMARKED
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) \(5x3y=1\) > \(5x=3y+1\). So \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are coprime, 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.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Director
Status: Apply  Last Chance
Affiliations: IIT, Purdue, PhD, TauBetaPi
Joined: 18 Jul 2010
Posts: 685
Schools: Wharton, Sloan, Chicago, Haas
WE 1: 8 years in Oil&Gas

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
23 Aug 2010, 16:01
Bunuel, Is there a complete discussion on GCDs and LCMs on the forum? Can you please point me to the same? I am trying to recollect why is x y = GCD(x,y) x LCM(x,y)? Thanks
_________________
Consider kudos, they are good for health



Moderator
Joined: 01 Sep 2010
Posts: 3211

If x and y are positive integers, what is the greatest [#permalink]
Show Tags
04 Mar 2012, 07:21
14
This post received KUDOS
35
This post was BOOKMARKED
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 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......
_________________
COLLECTION OF QUESTIONS AND RESOURCES Quant: 1. ALL GMATPrep questions Quant/Verbal 2. Bunuel Signature Collection  The Next Generation 3. Bunuel Signature Collection ALLINONE WITH SOLUTIONS 4. Veritas Prep Blog PDF Version 5. MGMAT Study Hall Thursdays with Ron Quant Videos Verbal:1. Verbal question bank and directories by Carcass 2. MGMAT Study Hall Thursdays with Ron Verbal Videos 3. Critical Reasoning_Oldy but goldy question banks 4. Sentence Correction_Oldy but goldy question banks 5. Readingcomprehension_Oldy but goldy question banks
Last edited by Bunuel on 09 Jul 2014, 10:50, edited 2 times in total.
Edited the question and the OA



Math Expert
Joined: 02 Sep 2009
Posts: 39662

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
04 Mar 2012, 07:41
32
This post received KUDOS
Expert's post
29
This post was BOOKMARKED
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) \(5x3y=1\) > \(5x=3y+1\) > \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are coprime, 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.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Moderator
Joined: 01 Sep 2010
Posts: 3211

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
04 Mar 2012, 09:22



Math Expert
Joined: 02 Sep 2009
Posts: 39662

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
05 Mar 2012, 02:28



Intern
Joined: 30 Jan 2012
Posts: 2

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
07 Apr 2012, 15:55
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) \(5x3y=1\) > \(5x=3y+1\) > \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are coprime, 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.



Manager
Joined: 17 Apr 2012
Posts: 72
GMAT Date: 11022012

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
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) \(5x3y=1\) > \(5x=3y+1\) > \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are coprime, 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,



Manager
Joined: 09 Apr 2012
Posts: 62

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
05 Sep 2012, 14:04
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.



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7440
Location: Pune, India

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
05 Sep 2012, 22:05
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 Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Intern
Joined: 23 Sep 2008
Posts: 24

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
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=732x 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) 5x3y=1=> y=(5x1)/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



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7440
Location: Pune, India

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
16 Sep 2012, 23:58
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=732x 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) 5x3y=1=> y=(5x1)/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.
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Manager
Joined: 04 Jan 2014
Posts: 123

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
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) \(5x3y=1\) > \(5x=3y+1\) > \(5x\) and \(3y\) are consecutive integers. Two consecutive integers are coprime, 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!



Math Expert
Joined: 02 Sep 2009
Posts: 39662

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
18 May 2014, 01:29



Senior Manager
Joined: 28 Apr 2014
Posts: 282

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
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 ?



Math Expert
Joined: 02 Sep 2009
Posts: 39662

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
07 Jul 2014, 01:12
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.
_________________
New to the Math Forum? Please read this: All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Senior Manager
Joined: 28 Apr 2014
Posts: 282

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
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
Sent from my iPhone using Tapatalk



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7440
Location: Pune, India

Re: If x and y are positive integers, what is the greatest [#permalink]
Show Tags
07 Jul 2014, 06:02
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
Sent from my iPhone using Tapatalk 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 34 questions, all questions are fair play.
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews




Re: If x and y are positive integers, what is the greatest
[#permalink]
07 Jul 2014, 06:02



Go to page
1 2 3
Next
[ 48 posts ]





Similar topics 
Author 
Replies 
Last post 
Similar Topics:




What is the Greatest Common divisor of two positive integers x and y?

stonecold 
3 
27 Nov 2016, 04:38 

1


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

alanforde800Maximus 
1 
25 Oct 2016, 10:55 

2


If x and y are both positive integers, what is their greatest common

Bunuel 
8 
25 Apr 2017, 15:37 

3


What is the greatest common factor of the positive integers x and y? 1

Bunuel 
6 
18 Apr 2016, 07:22 

19


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

shasadou 
3 
13 Jan 2017, 01:03 



