Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 21 Jul 2009
Posts: 11

If a, b, and c are positive integers, with a < b < c, [#permalink]
Show Tags
22 Oct 2009, 23:27
1
This post received KUDOS
9
This post was BOOKMARKED
Question Stats:
58% (02:49) correct
42% (01:55) wrong based on 355 sessions
HideShow timer Statistics
If a, b, and c are positive integers, with a < b < c, are a, b, and c consecutive integers? (1) 1/a – 1/b = 1/c (2) a + c = b^2 – 1
Official Answer and Stats are available only to registered users. Register/ Login.
Last edited by Bunuel on 10 Jun 2012, 23:02, edited 1 time in total.
Edited the question.



Math Expert
Joined: 02 Sep 2009
Posts: 39662

Re: are a, b, and c consecutive integers? [#permalink]
Show Tags
22 Oct 2009, 23:59
3
This post received KUDOS
Expert's post
7
This post was BOOKMARKED
SMAbbas wrote: If a, b, and c are positive integers, with a < b < c, are a, b, and c consecutive integers? (1) 1/a – 1/b = 1/c (2) a + c = b2 – 1 IMO D, Please verify my answer If \(a\), \(b\), \(c\) are consecutive and \(a<b<c\), then must be true that: \(b=a+1\) and \(c=a+2\). (1) \(\frac{1}{a}\frac{1}{b}=\frac{1}{c}\) > \((ba)c=ab\) > \((a+1a)(a+2)=a(a+1)\) > \(a^2=2\) > \(a^2=2\) as \(a\) is positive integer this equation has no positive integer roots > \(a\), \(b\), \(c\), are not consecutive positive integers. Sufficient. (2) \(a+c=b^21\) > \(a+a+2=(a+1)^21\) > \(2a+2=a^2+2a+11\) > \(a^2=2\). The same here: \(a\), \(b\), \(c\), are not consecutive positive integers. Sufficient. Answer D.
_________________
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



Current Student
Joined: 21 Oct 2009
Posts: 41

Re: are a, b, and c consecutive integers? [#permalink]
Show Tags
27 Oct 2009, 07:01
I think I must have misunderstood the basic rule on DS
Like this question, when we find it SUFFICIENT to say "it can't be a real number," it should be good enought to choose (D).
I thought I could choose (D) as long as I could verify YES for quesitions.



Senior Manager
Affiliations: SPG
Joined: 15 Nov 2006
Posts: 327

Re: are a, b, and c consecutive integers? [#permalink]
Show Tags
22 May 2010, 08:19
1
This post received KUDOS
i approached the question differently ... this may not be the best way but i'm going to share it anyway ... sharing is caring
a, b, c > 0 [positive integers] b=a+1, c=a+2 [a, b, c are consecutive integers]
(1) 1/a  1/b = 1/c 1/a  1/a+1 = 1/a+2 1/a(a+1) = 1/(a+2) a+2 = a(a+1) c = a*b [c=a+2, b=a+1]
if a, b, c are consecutive ... the above equation in bold can never be true 2 ? 0*1 3 ? 1*2 4 ? 2*3
1 is sufficient to explain that a, b, c are not consecutive integers
(2) a+c = (b1)(b+1) a+c = a*c [a=b1, c=b+1]
if a, b, c are consecutive ... the above equation in bold can never be true 0+2 ? 0*2 1+3 ? 1*3 2+4 ? 2*4
2 is sufficient to explain that a, b, c are not consecutive integers
ans is D
_________________
press kudos, if you like the explanation, appreciate the effort or encourage people to respond.
Download the Ultimate SC Flashcards



Senior Manager
Joined: 19 Nov 2009
Posts: 320

Re: are a, b, and c consecutive integers? [#permalink]
Show Tags
22 May 2010, 16:01
i approached the question differently ... this may not be the best way but i'm going to share it anyway ... sharing is caring As long as you get the right answer, any approach should be fine !



Manager
Joined: 02 Jan 2010
Posts: 134

Re: are a, b, and c consecutive integers? [#permalink]
Show Tags
22 May 2010, 20:43
I got D too, used sample numbers to check. The methods given here are useful as well!!!
_________________
Regards Ganesh Class of 2012 Great Lakes Institute of Management http://greatlakes.edu.in



Manager
Joined: 29 Jun 2011
Posts: 161
WE 1: Information Technology(Retail)

Re: If a, b, and c are positive integers, with a < b < c, [#permalink]
Show Tags
03 Sep 2013, 00:11
I solved the questions by plugging in consecutive numbers. Got the ans D. Would you suggest to use plugging in method to solve such questions?



Current Student
Joined: 06 Sep 2013
Posts: 1997
Concentration: Finance

Re: If a, b, and c are positive integers, with a < b < c, [#permalink]
Show Tags
24 Oct 2013, 07:05
1
This post received KUDOS
SMAbbas wrote: If a, b, and c are positive integers, with a < b < c, are a, b, and c consecutive integers?
(1) 1/a – 1/b = 1/c
(2) a + c = b^2 – 1 Hey guys  The way I did it was (1) 1/a – 1/b = 1/c > 1/a = 1/b + 1/c LCM of 'b' and 'c' would never be smaller than b or c because these numbers are coprime which means that they do not share any common integer factor other than 1. Now since c>b>a then they following the logic above they can't be consecutive integers. Sufficient (2) a + c = b^2 – 1 > a+c = (b+1)(b1) Now if a,b,c were consecutive integers it will be true that (a+c)/2 is equal to b So we would have that a+c = 2b Now if we replace a+c = 2b in the first equation we would have that 2b = b^21 And when trying to solve this quadratic we would realize that b is not an integer, since we would have to use the formula So it contradicts the information that a,b,c are integers. Hence a,b,c cannot be consecutive integers Insuff Answer (D) Please let me know whether this makes sense. If it does, throw me some Kudos! Cheers J
Last edited by jlgdr on 24 Oct 2013, 08:05, edited 1 time in total.



Math Expert
Joined: 02 Sep 2009
Posts: 39662

Re: If a, b, and c are positive integers, with a < b < c, [#permalink]
Show Tags
24 Oct 2013, 07:41
jlgdr wrote: SMAbbas wrote: If a, b, and c are positive integers, with a < b < c, are a, b, and c consecutive integers?
(1) 1/a – 1/b = 1/c
(2) a + c = b^2 – 1 Hey guys  The way I did it was (1) 1/a – 1/b = 1/c > 1/a = 1/b + 1/c LCM of 'b' and 'c' would never be smaller than b or c because these numbers are coprime which means that they do not share any common integer factor other than 1. Now since c>b>a then they following the logic above they can't be consecutive integers. Sufficient (2) a + c = b^2 – 1 > a+c = (b+1)(b1) Now if a,b,c were consecutive integers it will be true that (a+c)/2 is equal to b So we would have that a+c = 2b Now if we replace a+c = 2b in the first equation we would have that 2b = b^21 And when trying to solve this quadratic we would realize that b is not an integer, since we would have to use the formula So it contradicts the information that a,b,c are integers. Hence a,b,c cannot be consecutive integers Insuff
Answer (E)Please let me know whether this makes sense. If it does, throw me some Kudos! Cheers J Notice that OA is D.
_________________
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



Current Student
Joined: 06 Sep 2013
Posts: 1997
Concentration: Finance

Re: If a, b, and c are positive integers, with a < b < c, [#permalink]
Show Tags
24 Oct 2013, 08:06
Bunuel wrote: jlgdr wrote: SMAbbas wrote: If a, b, and c are positive integers, with a < b < c, are a, b, and c consecutive integers?
(1) 1/a – 1/b = 1/c
(2) a + c = b^2 – 1 Hey guys  The way I did it was (1) 1/a – 1/b = 1/c > 1/a = 1/b + 1/c LCM of 'b' and 'c' would never be smaller than b or c because these numbers are coprime which means that they do not share any common integer factor other than 1. Now since c>b>a then they following the logic above they can't be consecutive integers. Sufficient (2) a + c = b^2 – 1 > a+c = (b+1)(b1) Now if a,b,c were consecutive integers it will be true that (a+c)/2 is equal to b So we would have that a+c = 2b Now if we replace a+c = 2b in the first equation we would have that 2b = b^21 And when trying to solve this quadratic we would realize that b is not an integer, since we would have to use the formula So it contradicts the information that a,b,c are integers. Hence a,b,c cannot be consecutive integers Insuff
Answer (E)Please let me know whether this makes sense. If it does, throw me some Kudos! Cheers J Notice that OA is D. Oops, typo. Yes, answer is (D) indeed



Senior Manager
Joined: 23 Oct 2010
Posts: 383
Location: Azerbaijan
Concentration: Finance

Re: If a, b, and c are positive integers, with a < b < c, [#permalink]
Show Tags
24 Oct 2013, 21:53
1) 1/a – 1/b = 1/c let a=1 b=2 c=3 1/21/3 doesnt equal to 1/4 (2) a + c = b^2 – 1 let a(even) b(odd) c(even) a + c = b^2 – 1 = even +even =odd^1=even  ok! let a(odd) b(even) c(odd) a + c = b^2 – 1 = odd +odd doesnt equal to even^1 not ok!
_________________
Happy are those who dream dreams and are ready to pay the price to make them come true
I am still on all gmat forums. msg me if you want to ask me smth



Manager
Joined: 25 Oct 2013
Posts: 169

Re: If a, b, and c are positive integers, with a < b < c, [#permalink]
Show Tags
04 Feb 2014, 06:24
I assumed a,b,c to be consecutive and substituted b1,b,b+1 in each statement independently. (1) gives b^2=2b+1 and b is not an integer hence my assumption that a,b,c are consecutive cannot be true. sufficient.(2) also gives b^2 = 2b+1. same as (1) hence sufficient.+1 D. Hope it made sense.
_________________
Click on Kudos if you liked the post!
Practice makes Perfect.



Intern
Joined: 05 Feb 2014
Posts: 48

Re: are a, b, and c consecutive integers? [#permalink]
Show Tags
14 May 2014, 04:40
Bunuel wrote: SMAbbas wrote: If a, b, and c are positive integers, with a < b < c, are a, b, and c consecutive integers? (1) 1/a – 1/b = 1/c (2) a + c = b2 – 1 IMO D, Please verify my answer If \(a\), \(b\), \(c\) are consecutive and \(a<b<c\), then must be true that: \(b=a+1\) and \(c=a+2\). (1) \(\frac{1}{a}\frac{1}{b}=\frac{1}{c}\) > \((ba)c=ab\) > \((a+1a)(a+2)=a(a+1)\) > \(a^2=2\) > \(a^2=2\) as \(a\) is positive integer this equation has no positive integer roots > \(a\), \(b\), \(c\), are not consecutive positive integers. Sufficient. (2) \(a+c=b^21\) > \(a+a+2=(a+1)^21\) > \(2a+2=a^2+2a+11\) > \(a^2=2\). The same here: \(a\), \(b\), \(c\), are not consecutive positive integers. Sufficient. Answer D. Hi Bunuel, Thanks for the answer. One question  what is the meaning of "this eq has no positive integer root ?". It is because the eq has A = + 2 ?



Math Expert
Joined: 02 Sep 2009
Posts: 39662

Re: are a, b, and c consecutive integers? [#permalink]
Show Tags
14 May 2014, 06:35
gauravsoni wrote: Bunuel wrote: SMAbbas wrote: If a, b, and c are positive integers, with a < b < c, are a, b, and c consecutive integers? (1) 1/a – 1/b = 1/c (2) a + c = b2 – 1 IMO D, Please verify my answer If \(a\), \(b\), \(c\) are consecutive and \(a<b<c\), then must be true that: \(b=a+1\) and \(c=a+2\). (1) \(\frac{1}{a}\frac{1}{b}=\frac{1}{c}\) > \((ba)c=ab\) > \((a+1a)(a+2)=a(a+1)\) > \(a^2=2\) > \(a^2=2\) as \(a\) is positive integer this equation has no positive integer roots > \(a\), \(b\), \(c\), are not consecutive positive integers. Sufficient. (2) \(a+c=b^21\) > \(a+a+2=(a+1)^21\) > \(2a+2=a^2+2a+11\) > \(a^2=2\). The same here: \(a\), \(b\), \(c\), are not consecutive positive integers. Sufficient. Answer D. Hi Bunuel, Thanks for the answer. One question  what is the meaning of "this eq has no positive integer root ?". It is because the eq has A = + 2 ? We are given that \(a\) is a positive integer, while from \(a^2=2\), \(a=\sqrt{2}\) or \(a=\sqrt{2}\). Hence our assumption that a, b, and c were consecutive integers was wrong.
_________________
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



Intern
Joined: 05 Feb 2014
Posts: 48

Re: are a, b, and c consecutive integers? [#permalink]
Show Tags
14 May 2014, 07:29
gauravsoni wrote: Bunuel wrote: SMAbbas wrote: If a, b, and c are positive integers, with a < b < c, are a, b, and c consecutive integers? (1) 1/a – 1/b = 1/c (2) a + c = b2 – 1 IMO D, Please verify my answer If \(a\), \(b\), \(c\) are consecutive and \(a<b<c\), then must be true that: \(b=a+1\) and \(c=a+2\). (1) \(\frac{1}{a}\frac{1}{b}=\frac{1}{c}\) > \((ba)c=ab\) > \((a+1a)(a+2)=a(a+1)\) > \(a^2=2\) > \(a^2=2\) as \(a\) is positive integer this equation has no positive integer roots > \(a\), \(b\), \(c\), are not consecutive positive integers. Sufficient. (2) \(a+c=b^21\) > \(a+a+2=(a+1)^21\) > \(2a+2=a^2+2a+11\) > \(a^2=2\). The same here: \(a\), \(b\), \(c\), are not consecutive positive integers. Sufficient. Answer D. Hi Bunuel, Thanks for the answer. One question  what is the meaning of "this eq has no positive integer root ?". It is because the eq has A = + 2 ? We are given that \(a\) is a positive integer, while from \(a^2=2\), \(a=\sqrt{2}\) or \(a=\sqrt{2}\). Hence our assumption that a, b, and c were consecutive integers was wrong.[/quote] Got it , thanks...



Intern
Joined: 25 Jan 2014
Posts: 46
Concentration: Strategy, International Business
GMAT 1: 600 Q44 V29 GMAT 2: 710 Q48 V38
GPA: 3.35
WE: Analyst (Computer Software)

Re: If a, b, and c are positive integers, with a < b < c, [#permalink]
Show Tags
15 May 2014, 03:32
Bunuel,
We can also do it by taking a, b and c as (x1), x, (x+1) right? It looks the same to me as taking b = a+1 and c = a+2



Math Expert
Joined: 02 Sep 2009
Posts: 39662

Re: If a, b, and c are positive integers, with a < b < c, [#permalink]
Show Tags
15 May 2014, 03:42



Intern
Joined: 25 Jan 2014
Posts: 46
Concentration: Strategy, International Business
GMAT 1: 600 Q44 V29 GMAT 2: 710 Q48 V38
GPA: 3.35
WE: Analyst (Computer Software)

Re: If a, b, and c are positive integers, with a < b < c, [#permalink]
Show Tags
15 May 2014, 03:46
Thanks a lot Bunuel



GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15958

Re: If a, b, and c are positive integers, with a < b < c, [#permalink]
Show Tags
28 Jun 2015, 04:05
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources



CEO
Joined: 17 Jul 2014
Posts: 2524
Location: United States (IL)
Concentration: Finance, Economics
GPA: 3.92
WE: General Management (Transportation)

If a, b, and c are positive integers, with a < b < c, [#permalink]
Show Tags
02 Feb 2016, 19:08
took me 2 mins to crack this one.. if integers are consecutive, then we can write as: a a+1 a+2
we have: 1/a  1/a+1 = a+1a/a(a+1) = 1/a^2+1 a^2+1 if a=1, then c=2, impossible. if a=2, then c=5, again impossible. we can see that neither of the options works here.
2. a+c=b^2 +1 a+a+2 = (a+1)^2 +1 2a+2 = a^2+2a a^2=2 this is impossible, since all the numbers are positive, and a must be > than 0 and an integer. we can see that we have a definite answer, and the answer is D.




If a, b, and c are positive integers, with a < b < c,
[#permalink]
02 Feb 2016, 19:08








Similar topics 
Author 
Replies 
Last post 
Similar Topics:


10


If a, b, and c are positive integers such that a < b < c, is a% of b%

Bunuel 
11 
11 Apr 2017, 22:01 

6


If a, b, and c are positive integers, with a < b < c, are a, b, and c

Bunuel 
5 
19 Nov 2014, 07:44 

1


A, B, C and D are positive integers, and A < B < C < D. If A

honchos 
2 
25 Aug 2014, 02:49 



If a, b, and c are positive numbers, is a < b < c?

Smita04 
2 
14 Apr 2012, 01:45 

4


a, b, c are consecutive integers, so that a < b < c. Is the

eladshush 
5 
16 Jun 2016, 16:09 



