January 17, 2019 January 17, 2019 08:00 AM PST 09:00 AM PST Learn the winning strategy for a high GRE score — what do people who reach a high score do differently? We're going to share insights, tips and strategies from data we've collected from over 50,000 students who used examPAL. January 19, 2019 January 19, 2019 07:00 AM PST 09:00 AM PST Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT.
Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 52142

Question Stats:
55% (01:44) correct 45% (01:45) wrong based on 82 sessions
HideShow timer Statistics



Math Expert
Joined: 02 Sep 2009
Posts: 52142

Re M3131
[#permalink]
Show Tags
14 Jun 2015, 13:27
Official Solution:If \(x\) is an integer, what is the remainder when \(1  x^2\) is divided by 4? Notice that if \(x\) is odd, then \(1  x^2\) is a multiple of 4. For example: If \(x=1\), \(1  x^2 = 0\); If \(x=3\), \(1  x^2 = 8\); If \(x=5\), \(1  x^2 = 24\). ... (1) The sum of any two factors of \(x\) is even. For the sum of ANY two factors of \(x\) to be even all factors of \(x\) must be odd (even if one of the factors is even then we could pair that even factor with 1, which is a factor of every integer, and we'd get odd sum), which means that \(x\) is an odd number. Sufficient. (2) The product of any two factors of \(x\) is odd. Basically the same here: for the product of ANY two factors of \(x\) to be odd all factors of \(x\) must be odd (even if one of the factors is even then we could pair that even factor with any other factor and we'd get even product), which means that \(x\) is an odd number. Sufficient. Answer: D
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  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: 25 Apr 2015
Posts: 10

Re: M3131
[#permalink]
Show Tags
31 Oct 2015, 06:07
Bunuel wrote: Official Solution:
If \(x\) is an integer, what is the remainder when \(1  x^2\) is divided by 4? Notice that if \(x\) is odd, then \(1  x^2\) is a multiple of 4. For example: If \(x=1\), \(1  x^2 = 0\); If \(x=3\), \(1  x^2 = 8\); If \(x=5\), \(1  x^2 = 24\). ... (1) The sum of any two factors of \(x\) is even. For the sum of ANY two factors of \(x\) to be even all factors of \(x\) must be odd (even if one of the factors is even then we could pair that even factor with 1, which is a factor of every integer, and we'd get odd sum), which means that \(x\) is an odd number. Sufficient. (2) The product of any two factors of \(x\) is odd. Basically the same here: for the product of ANY two factors of \(x\) to be odd all factors of \(x\) must be odd (even if one of the factors is even then we could pair that even factor with any other factor and we'd get even product), which means that \(x\) is an odd number. Sufficient.
Answer: D >> even if one of the factors is even then we could pair that even factor with 1, which is a factor of every integer, and we'd get odd sum), which means that \(x\) is an odd number Could you please give some example to illustrate the point? I am having difficulty comprehending this point.



Intern
Joined: 19 Dec 2015
Posts: 28

Bunuel wrote: Official Solution:
If \(x\) is an integer, what is the remainder when \(1  x^2\) is divided by 4? Notice that if \(x\) is odd, then \(1  x^2\) is a multiple of 4. For example: If \(x=1\), \(1  x^2 = 0\); If \(x=3\), \(1  x^2 = 8\); If \(x=5\), \(1  x^2 = 24\). ... (1) The sum of any two factors of \(x\) is even. For the sum of ANY two factors of \(x\) to be even all factors of \(x\) must be odd (even if one of the factors is even then we could pair that even factor with 1, which is a factor of every integer, and we'd get odd sum), which means that \(x\) is an odd number. Sufficient. (2) The product of any two factors of \(x\) is odd. Basically the same here: for the product of ANY two factors of \(x\) to be odd all factors of \(x\) must be odd (even if one of the factors is even then we could pair that even factor with any other factor and we'd get even product), which means that \(x\) is an odd number. Sufficient.
Answer: D Bunuel, is this a correct line of reasoning for your explaination on statement 1?: If \(x = 4\), we have three factors; \(1, 2\) and \(2\). From S1, we get that the sum of ANY two factors of \(x = even\), which includes \(1\). So, we have two cases: 1) \(1 + 2 = 3 = odd\) 2) \(2 + 2 = 4 = even\) Thus, not all/ANY factors of 4, when added together, gives an even number. However, if \(x = 5\), we only have two factors; \(1\) and \(5\). This only gives one case: \(1 + 5 = 6 = even\). Since \(5\) only has these two factors, we will always have an even number when any two factors of \(5\) are added. Conclusion: \(x\) must be an odd number for ANY two factors of \(x\) to be \(even\). Side note, these are our even/odd rules for addition: 1) \(Even + even = even\) 2) \(Odd + odd = even\) 3) \(Even + odd = odd\) 4) \(Odd + even = odd\)



Math Expert
Joined: 02 Sep 2009
Posts: 52142

Re: M3131
[#permalink]
Show Tags
18 May 2016, 11:38
Sallyzodiac wrote: Bunuel wrote: Official Solution:
If \(x\) is an integer, what is the remainder when \(1  x^2\) is divided by 4? Notice that if \(x\) is odd, then \(1  x^2\) is a multiple of 4. For example: If \(x=1\), \(1  x^2 = 0\); If \(x=3\), \(1  x^2 = 8\); If \(x=5\), \(1  x^2 = 24\). ... (1) The sum of any two factors of \(x\) is even. For the sum of ANY two factors of \(x\) to be even all factors of \(x\) must be odd (even if one of the factors is even then we could pair that even factor with 1, which is a factor of every integer, and we'd get odd sum), which means that \(x\) is an odd number. Sufficient. (2) The product of any two factors of \(x\) is odd. Basically the same here: for the product of ANY two factors of \(x\) to be odd all factors of \(x\) must be odd (even if one of the factors is even then we could pair that even factor with any other factor and we'd get even product), which means that \(x\) is an odd number. Sufficient.
Answer: D Bunuel, is this a correct line of reasoning for your explaination on statement 1?: If \(x = 4\), we have three factors; \(1, 2\) and \(2\). From S1, we get that the sum of ANY two factors of \(x = even\), which includes \(1\). So, we have two cases: 1) \(1 + 2 = 3 = odd\) 2) \(2 + 2 = 4 = even\) Thus, not all/ANY factors of 4, when added together, gives an even number. However, if \(x = 5\), we only have two factors; \(1\) and \(5\). This only gives one case: \(1 + 5 = 6 = even\). Since \(5\) only has these two factors, we will always have an even number when any two factors of \(5\) are added. Conclusion: \(x\) must be an odd number for ANY two factors of \(x\) to be \(even\). Side note, these are our even/odd rules for addition: 1) \(Even + even = even\) 2) \(Odd + odd = even\) 3) \(Even + odd = odd\) 4) \(Odd + even = odd\) ______________ Yes that's correct.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  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: 31 Mar 2016
Posts: 384
Location: India
Concentration: Operations, Finance
GPA: 3.8
WE: Operations (Commercial Banking)

Re M3131
[#permalink]
Show Tags
08 Jul 2016, 08:02
I think this is a highquality question and the explanation isn't clear enough, please elaborate. For choice 1 to be sufficient, there is no constraint in the question that rules out considering 0 which is also an even number. Doesn't matter whether you state "any factors" but that does not necessarily rule out "0" which is an even number as well in which case remainder will be 0. Can you explain?



Math Expert
Joined: 02 Sep 2009
Posts: 52142

Re: M3131
[#permalink]
Show Tags
08 Jul 2016, 08:11



Retired Moderator
Joined: 26 Nov 2012
Posts: 591

Re: M3131
[#permalink]
Show Tags
16 May 2017, 09:59
Hi Bunuel,
My analysis is exactly same as yours. But I chose the option E. Since we are not getting a unique odd number. We are just concluding that x is an odd from both the statements. From Stat 1, x could be 1 and from stat 2 x could be 3 and the mentioned numbers satisfy the given respective statements.
I think you have marked D as the correct answer since GMAT doesn't contradict with the statement i.e. two different cases can't be correct. ?
Can you please throw some light on this.



Math Expert
Joined: 02 Sep 2009
Posts: 52142

Re: M3131
[#permalink]
Show Tags
16 May 2017, 10:06
msk0657 wrote: Hi Bunuel,
My analysis is exactly same as yours. But I chose the option E. Since we are not getting a unique odd number. We are just concluding that x is an odd from both the statements. From Stat 1, x could be 1 and from stat 2 x could be 3 and the mentioned numbers satisfy the given respective statements.
I think you have marked D as the correct answer since GMAT doesn't contradict with the statement i.e. two different cases can't be correct. ?
Can you please throw some light on this. The question asks: what is the remainder when 1−x^2 is divided by 4? NOT what is the value of x. We concluded that if x is odd number (any odd number), then the reminder when 1−x^2 is divided by 4 is 0. From each statement we got that x must be odd, thus from each statement we have that the remainder must be 0. Therefore each statement gives unique answer to the question (0) and thus each is sufficient. Answer D. Hope it's clear.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  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: 19 Aug 2016
Posts: 149
Location: India
GPA: 3.82

Re: M3131
[#permalink]
Show Tags
17 Jul 2017, 08:42
Bunuel wrote: Official Solution:
If \(x\) is an integer, what is the remainder when \(1  x^2\) is divided by 4? Notice that if \(x\) is odd, then \(1  x^2\) is a multiple of 4. For example: If \(x=1\), \(1  x^2 = 0\); If \(x=3\), \(1  x^2 = 8\); If \(x=5\), \(1  x^2 = 24\). ... (1) The sum of any two factors of \(x\) is even. For the sum of ANY two factors of \(x\) to be even all factors of \(x\) must be odd (even if one of the factors is even then we could pair that even factor with 1, which is a factor of every integer, and we'd get odd sum), which means that \(x\) is an odd number. Sufficient.
Answer: D Hi, The line of reasoning given in the first statement is that becuase sum of any two factors of x is even, all the factors of x should be odd. Why are we assuming this? It could also be that all the factors of x is even, in that case also the sum of any two factors of x would be even as well, right? Secondly, I did not understand the '(even if one of the factors is even then we could pair that even factor with 1, which is a factor of every integer, and we'd get odd sum)' part. If we are getting one pair of sum as odd, isn't that rendering the statement untrue? Can someone explain an alternate method of explaining statement 1? Thanks!
_________________
Consider giving me Kudos if you find my posts useful, challenging and helpful!



Math Expert
Joined: 02 Sep 2009
Posts: 52142

Re: M3131
[#permalink]
Show Tags
17 Jul 2017, 08:47
ashikaverma13 wrote: Bunuel wrote: Official Solution:
If \(x\) is an integer, what is the remainder when \(1  x^2\) is divided by 4? Notice that if \(x\) is odd, then \(1  x^2\) is a multiple of 4. For example: If \(x=1\), \(1  x^2 = 0\); If \(x=3\), \(1  x^2 = 8\); If \(x=5\), \(1  x^2 = 24\). ... (1) The sum of any two factors of \(x\) is even. For the sum of ANY two factors of \(x\) to be even all factors of \(x\) must be odd (even if one of the factors is even then we could pair that even factor with 1, which is a factor of every integer, and we'd get odd sum), which means that \(x\) is an odd number. Sufficient.
Answer: D Hi, The line of reasoning given in the first statement is that becuase sum of any two factors of x is even, all the factors of x should be odd. Why are we assuming this? It could also be that all the factors of x is even, in that case also the sum of any two factors of x would be even as well, right? Secondly, I did not understand the '(even if one of the factors is even then we could pair that even factor with 1, which is a factor of every integer, and we'd get odd sum)' part. If we are getting one pair of sum as odd, isn't that rendering the statement untrue? Can someone explain an alternate method of explaining statement 1? Thanks! An integer cannot have only even factors because 1, an odd integer, is a factor of all integers. About you second point: exactly, the fact that even one even factor will make the statement untrue means that x cannot have any even factors. Hope it's clear.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  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: 19 Aug 2016
Posts: 149
Location: India
GPA: 3.82

Re: M3131
[#permalink]
Show Tags
17 Jul 2017, 11:21
Bunuel wrote: ashikaverma13 wrote: Bunuel wrote: Official Solution:
If \(x\) is an integer, what is the remainder when \(1  x^2\) is divided by 4? Notice that if \(x\) is odd, then \(1  x^2\) is a multiple of 4. For example: If \(x=1\), \(1  x^2 = 0\); If \(x=3\), \(1  x^2 = 8\); If \(x=5\), \(1  x^2 = 24\). ... (1) The sum of any two factors of \(x\) is even. For the sum of ANY two factors of \(x\) to be even all factors of \(x\) must be odd (even if one of the factors is even then we could pair that even factor with 1, which is a factor of every integer, and we'd get odd sum), which means that \(x\) is an odd number. Sufficient.
Answer: D Hi, The line of reasoning given in the first statement is that becuase sum of any two factors of x is even, all the factors of x should be odd. Why are we assuming this? It could also be that all the factors of x is even, in that case also the sum of any two factors of x would be even as well, right? Secondly, I did not understand the '(even if one of the factors is even then we could pair that even factor with 1, which is a factor of every integer, and we'd get odd sum)' part. If we are getting one pair of sum as odd, isn't that rendering the statement untrue? Can someone explain an alternate method of explaining statement 1? Thanks! An integer cannot have only even factors because 1, an odd integer, is a factor of all integers. About you second point: exactly, the fact that even one even factor will make the statement untrue means that x cannot have any even factors. Hope it's clear. yes, I think I understand now. Thanks
_________________
Consider giving me Kudos if you find my posts useful, challenging and helpful!



Manager
Joined: 02 Nov 2015
Posts: 165

what if I consider the factors of 21 as 3 and 7 and then proceed for option 2. its 121^2, which yields 440 leading us to a remainder of 0. where am I missing. Pls help..



Math Expert
Joined: 02 Sep 2009
Posts: 52142

Re: M3131
[#permalink]
Show Tags
18 Jul 2017, 08:16



Manager
Joined: 02 Nov 2015
Posts: 165

Re: M3131
[#permalink]
Show Tags
18 Jul 2017, 08:49
Yes yes ..... My bad. My wrong interpretation. Thanks Bunuel !!!!!



Current Student
Joined: 03 Aug 2016
Posts: 353
Location: Canada
GMAT 1: 660 Q44 V38 GMAT 2: 690 Q46 V40
GPA: 3.9
WE: Information Technology (Consumer Products)

Re: M3131
[#permalink]
Show Tags
02 Oct 2017, 09:51
Bunuel . Vyshak What if x is 10 ? Factors 1,2,5,10. 5*1 = 5 is odd but remainder will be 1 1x*x = 1100 = 99/4
_________________
My MBA Journey  https://smalldoubledouble.com



Math Expert
Joined: 02 Sep 2009
Posts: 52142

Re: M3131
[#permalink]
Show Tags
02 Oct 2017, 09:55



Current Student
Joined: 03 Aug 2016
Posts: 353
Location: Canada
GMAT 1: 660 Q44 V38 GMAT 2: 690 Q46 V40
GPA: 3.9
WE: Information Technology (Consumer Products)

Re: M3131
[#permalink]
Show Tags
02 Oct 2017, 10:00
Bunuel wrote: mbsingh wrote: Bunuel . Vyshak What if x is 10 ? Factors 1,2,5,10. 5*1 = 5 is odd but remainder will be 1 1x*x = 1100 = 99/4 x cannot be 10 for neither of the statements. Please read above for more. Hi Bunuel, i don't understand why x cannot be 10. Is it because when statement 2 says product of any two factors is odd, does that imply that thats the only possible product for x's factors ?
_________________
My MBA Journey  https://smalldoubledouble.com



Math Expert
Joined: 02 Sep 2009
Posts: 52142

Re: M3131
[#permalink]
Show Tags
02 Oct 2017, 10:34
mbsingh wrote: Bunuel wrote: mbsingh wrote: Bunuel . Vyshak What if x is 10 ? Factors 1,2,5,10. 5*1 = 5 is odd but remainder will be 1 1x*x = 1100 = 99/4 x cannot be 10 for neither of the statements. Please read above for more. Hi Bunuel, i don't understand why x cannot be 10. Is it because when statement 2 says product of any two factors is odd, does that imply that thats the only possible product for x's factors ? The factor of 10 are 1, 2, 5, and 10. (1) says: The sum of ANY two factors of \(x\) is even. This is not true for 10. For example, 1 + 2 = 3 = odd. (2) The product of ANY two factors of \(x\) is odd. This is not true for 10. For example, 1*2 = 2 = even.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  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



Manager
Joined: 28 Jun 2018
Posts: 87
GMAT 1: 490 Q39 V18 GMAT 2: 640 Q47 V30 GMAT 3: 670 Q50 V31 GMAT 4: 700 Q49 V36
GPA: 4

Re: M3131
[#permalink]
Show Tags
30 Oct 2018, 00:15
BunuelPlease correct me if im wrong. We can also conclude from each statement that x is an odd PRIME number correct?







Go to page
1 2
Next
[ 23 posts ]



