Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 04 Aug 2010
Posts: 14

If n is a positive integer and r is the remainder when (n1) [#permalink]
Show Tags
04 Aug 2010, 03:47
10
This post was BOOKMARKED
Question Stats:
64% (02:37) correct
36% (01:39) wrong based on 265 sessions
HideShow timer Statistics
If n is a positive integer and r is the remainder when (n1)(n+1) is divided by 24, what is the value of r? (1) 2 is not a factor of n (2) 3 is not a factor of n
Official Answer and Stats are available only to registered users. Register/ Login.



Math Expert
Joined: 02 Sep 2009
Posts: 39755

Re: Data Sufficiency with remainder [#permalink]
Show Tags
04 Aug 2010, 03:54
3
This post received KUDOS
Expert's post
3
This post was BOOKMARKED
kwhitejr wrote: Can anyone demonstrate the following?
If n is a positive integer and r is the remainder when (n1)(n+1) is divided by 24, what is the value of r?
(1) 2 is not a factor of n (2) 3 is not a factor of n Hi, and welcome to Gmat Club! Below is the solution for your problem: If n is a positive integer and r is the remainder when (n1)(n+1) is divided by 24, what is the value of r?Number plugging method:\((n1)(n+1)=n^21\) (1) n is not divisible by 2 > pick two odd numbers: let's say 1 and 3 > if \(n=1\), then \(n^21=0\) and as zero is divisible by 24 (zero is divisible by any integer except zero itself) so remainder is 0 but if \(n=3\), then \(n^21=8\) and 8 divided by 24 yields remainder of 8. Two different answers, hence not sufficient. (2) n is not divisible by 3 > pick two numbers which are not divisible by 3: let's say 1 and 2 > if \(n=1\), then \(n^21=0\), so remainder is 0 but if \(n=2\), then \(n^21=3\) and 3 divided by 24 yields remainder of 3. Two different answers, hence not sufficient. (1)+(2) Let's check for several numbers which are not divisible by 2 or 3: \(n=1\) > \(n^21=0\) > remainder 0; \(n=5\) > \(n^21=24\) > remainder 0; \(n=7\) > \(n^21=48\) > remainder 0; \(n=11\) > \(n^21=120\) > remainder 0. Well it seems that all appropriate numbers will give remainder of 0. Sufficient. Algebraic approach:(1) n is not divisible by 2. Insufficient on its own, but this statement says that \(n=odd\) > \(n1\) and \(n+1\) are consecutive even integers > \((n1)(n+1)\) must be divisible by 8 (as both multiples are even and one of them will be divisible by 4. From consecutive even integers one is divisible by 4: (2, 4); (4, 6); (6, 8); (8, 10); (10, 12), ...). (2) n is not divisible by 3. Insufficient on its own, but form this statement either \(n1\) or \(n+1\) must be divisible by 3 (as \(n1\), \(n\), and \(n+1\) are consecutive integers, so one of them must be divisible by 3, we are told that it's not \(n\), hence either \(n1\) or \(n+1\)). (1)+(2) From (1) \((n1)(n+1)\) is divisible by 8, from (2) it's also divisible by 3, therefore it must be divisible by \(8*3=24\), which means that remainder upon division \((n1)(n+1)\) by 24 will be 0. Sufficient. Answer: C. Hope it helps.
_________________
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



Manager
Joined: 27 May 2010
Posts: 102

Re: Data Sufficiency with remainder [#permalink]
Show Tags
04 Aug 2010, 19:14
Thanks for the great explanation Bunuel.



Senior Manager
Status: Time to step up the tempo
Joined: 24 Jun 2010
Posts: 408
Location: Milky way
Schools: ISB, Tepper  CMU, Chicago Booth, LSB

Re: remainder [#permalink]
Show Tags
15 Aug 2010, 21:43
tt11234 wrote: hi all, please help solving the following question...
if n is a positive integer and r is the remainder when (n+1)(n1) is divided by 24. what is the value of r? 1) n is not divisible by 2 2) n is not divisible by 3 1) n is not divisible by 2 => n is not even. n could be 1, 3, 5, 7, 9 and (n+1) * (n1) could be different values. 2) n is not divisible by 3. n could be 1, 2, 4, 5, 7, 8, 10... again (n+1) * (n1) could be different values. Combine (1) and (2), n could be 1, 5, 7, 11, 13; ((n+1), (n1)) would contain a multiple of 6 and 4 when n is 5, 7, 11, 13 .. When n =1, (n1) is zero and the remainder is zero. Multiple of 6 and 4 would contain (3*2) and (2^2) = > (3 * 2 * 2 *2) = > 24. (n1)*(n+1) could be 4*6 or 6*8 and either way the remainder when (n1)*(n+1)/24 would be 0. Hence both statements is sufficient to answer this.
_________________
Support GMAT Club by putting a GMAT Club badge on your blog



Intern
Joined: 15 Aug 2010
Posts: 23
Location: Mumbai
Schools: Class of 2008, IIM Ahmedabad

Re: remainder [#permalink]
Show Tags
15 Aug 2010, 21:47
Statement 1 n is not divisible by 2. If n = 1, r = 0 If n = 3, r = 8. Hence, not sufficient. Statement 2 n is not divisible by 3 If n = 4, r = 15 If n = 5, r = 0 Hence, not sufficient. Combine both statements. n can be any of 1,5,7,11,13,17,19 etc. the respective r will always be 0. Hence, sufficient. PS: This way of solving would probably be better in a GMAT hall with a 2 minute time constraint. A rigorous algebraic proof can also be derived otherwise. Hope this helps. Thanks.
_________________
Naveenan Ramachandran 4GMAT  Mumbai



Manager
Joined: 17 May 2010
Posts: 121
Location: United States
Concentration: Entrepreneurship, Marketing
Schools: USC (Marshall)  Class of 2013
GPA: 3.26
WE: Brand Management (Consumer Products)

Re: Data Sufficiency with remainder [#permalink]
Show Tags
15 Sep 2010, 17:44
I got this problem right (on the GMATPrep #1 CAT) but I'm sure it took me at least 3 minutes, if not closer to 5. At least a minute to set up the problem and figure out how to approach it (the way described above is what I first thought of, but I tried to think of easier ways), another minute to test numbers, and another minute to think of ways on how my test could be wrong. Any suggestions on how to cut down on timing for unfamiliar problems like these? These divisor/remainder problems always get to me.
_________________
Discipline + Hard Work = Success! 770 (Q50, V46)



Manager
Joined: 30 Aug 2010
Posts: 91
Location: Bangalore, India

Re: Data Sufficiency with remainder [#permalink]
Show Tags
17 Sep 2010, 00:02
Folks, Consider this approach. Stmnt 1: N is not divisible by 2 ==> n=2k+1 ==> n1 * n+1 = 2k(2k+2)=4K(k+1) te K(K+1) is always divisible by 2 as they are consecutive integers. hence 4K(k+1) is always divisible by 8, but may not be by 24 (i.e. 8*3)....NOT Suff. Summary from stmnt 1: (n1)*(n+1) is always divisible by 8 Stmnt 2: N is not divisible by 3 ==> 2 cases > remainder can be 1 or 2 ==> n=3k+1 or 3K+2 with n=3K+1 ==> ==> n1 * n+1 = 3k(3k+2)=is divisible by 3 but not by 2 for ODD K: hence not divisible by 24 (i.e. 2^3 * 3) with n=3K+2 ==> ==> n1 * n+1 = (3k+1)(3k+3)=3(3k+1)(K+1) is divisible by 3 but not by 2 for EVEN K Not suff. Summary from stmnt 2: (n1)*(n+1) is always divisible by 3 1 & 2 summaries. (n1)*(n+1) is always divisible by 8*3 = 24..Suff. Answer "C" always divisible by 2 as they are consecutive integers. hence 4K(k+1) is always divisible by 8, but may not be by 24 (i.e. 8*3)....NOT Suff.



Intern
Joined: 09 Feb 2011
Posts: 11

Divisiblity remainder [#permalink]
Show Tags
11 Feb 2011, 00:37
If n is a positive integer and r is the remainder when (n – 1)(n + 1) is divided by 24, what is the value of r? (1) 2 is not a factor of n. (2) 3 is not a factor of
I marked ans e as we cannot find one value while using two other prime 5 & 7. So defainete value of r can be find out. But the ans is c. can anyone confim



Math Forum Moderator
Joined: 20 Dec 2010
Posts: 2010

Re: Divisiblity remainder [#permalink]
Show Tags
11 Feb 2011, 01:16
1
This post was BOOKMARKED
24 has the factors; 2*2*2*3 If (n1)(n+1) contains at least the above factors, it will be divisible by 24. 1. Means; n is odd; and n1 and n+1 are consecutive even; thus one of them will have at least 2 as factor and the other at least 2*2 Thus; we know (n1)(n+1) contains 2*2*2 as factor. But, it may or may not contain 3 as its factor. Not sufficient. 2. if n is not divisible by 3; one of (n1)(n+1) must be divisible by 3. One number of any set of 3 consecutive numbers is always divisible by 3. We know n is not divisible and n1,n,n+1 are consecutive. if n is not divisible by 3, one of (n1) and (n+1) must be divisible by 3 or have a factor as 3. However, we don't know whether these numbers have 2*2*2 as factors. Not sufficient. Combining both; (n1)(n+1) has factors 2*2*2*3 and must be divisible by 24. Remainder 0. Sufficient. Ans: "C"
_________________
~fluke
GMAT Club Premium Membership  big benefits and savings



Math Expert
Joined: 02 Sep 2009
Posts: 39755

Re: Divisiblity remainder [#permalink]
Show Tags
11 Feb 2011, 05:01



Director
Joined: 03 Aug 2012
Posts: 894
Concentration: General Management, General Management
GMAT 1: 630 Q47 V29 GMAT 2: 680 Q50 V32
GPA: 3.7
WE: Information Technology (Investment Banking)

Re: If n is a positive integer and r is the remainder when (n1) [#permalink]
Show Tags
15 Aug 2013, 21:27
1
This post received KUDOS
REM[ (n1)*(n+1)]/24 (1). 2 is not a factor of n => n is odd n=3 => (n1)*(n+1) = 2*4=8 => REM(8/24) =8 n=5 => (n1)*(n+1) = 4*6=24 => REM(24/24) =0 Hence INSUFFICIENT (2). 3 is not a factor of n n=2 => (n1)*(n+1) = 1*3=3 => REM(3/24) =3 n=4 => (n1)*(n+1) = 3*5=15 => REM(15/24) =15 Hence INSUFFICIENT Combining the two n=5,7,9,11,13 REM=0 Hence (C) it is
_________________
Rgds, TGC! _____________________________________________________________________ I Assisted You => KUDOS Please _____________________________________________________________________________



GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16030

Re: If n is a positive integer and r is the remainder when (n1) [#permalink]
Show Tags
02 Oct 2014, 09:06
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



Intern
Joined: 04 Nov 2012
Posts: 9

Re: If n is a positive integer and r is the remainder when (n1) [#permalink]
Show Tags
22 Feb 2015, 05:53
Bunuel wrote: kwhitejr wrote: Can anyone demonstrate the following?
If n is a positive integer and r is the remainder when (n1)(n+1) is divided by 24, what is the value of r?
(1) 2 is not a factor of n (2) 3 is not a factor of n Hi, and welcome to Gmat Club! Below is the solution for your problem: If n is a positive integer and r is the remainder when (n1)(n+1) is divided by 24, what is the value of r?Number plugging method:\((n1)(n+1)=n^21\) (1) n is not divisible by 2 > pick two odd numbers: let's say 1 and 3 > if \(n=1\), then \(n^21=0\) and as zero is divisible by 24 (zero is divisible by any integer except zero itself) so remainder is 0 but if \(n=3\), then \(n^21=8\) and 8 divided by 24 yields remainder of 8. Two different answers, hence not sufficient. (2) n is not divisible by 3 > pick two numbers which are not divisible by 3: let's say 1 and 2 > if \(n=1\), then \(n^21=0\), so remainder is 0 but if \(n=2\), then \(n^21=3\) and 3 divided by 24 yields remainder of 3. Two different answers, hence not sufficient. (1)+(2) Let's check for several numbers which are not divisible by 2 or 3: \(n=1\) > \(n^21=0\) > remainder 0; \(n=5\) > \(n^21=24\) > remainder 0; \(n=7\) > \(n^21=48\) > remainder 0; \(n=11\) > \(n^21=120\) > remainder 0. Well it seems that all appropriate numbers will give remainder of 0. Sufficient. Algebraic approach:(1) n is not divisible by 2. Insufficient on its own, but this statement says that \(n=odd\) > \(n1\) and \(n+1\) are consecutive even integers > \((n1)(n+1)\) must be divisible by 8 (as both multiples are even and one of them will be divisible by 4. From consecutive even integers one is divisible by 4: (2, 4); (4, 6); (6, 8); (8, 10); (10, 12), ...). (2) n is not divisible by 3. Insufficient on its own, but form this statement either \(n1\) or \(n+1\) must be divisible by 3 (as \(n1\), \(n\), and \(n+1\) are consecutive integers, so one of them must be divisible by 3, we are told that it's not \(n\), hence either \(n1\) or \(n+1\)). (1)+(2) From (1) \((n1)(n+1)\) is divisible by 8, from (2) it's also divisible by 3, therefore it must be divisible by \(8*3=24\), which means that remainder upon division \((n1)(n+1)\) by 24 will be 0. Sufficient. Answer: C. Hope it helps. Dear Bunuel, Thanks for sharing the alternatives to approaching this problem. I have a question here. What if we choose a number, n, such as 23 (which is neither divisible by 2 nor by 3)? In that case the given expression (n1)*(n+1) will be 22*24, which when divided by 24 will leave a remainder 22. Kindly advise! Thanks!



Math Expert
Joined: 02 Sep 2009
Posts: 39755

Re: If n is a positive integer and r is the remainder when (n1) [#permalink]
Show Tags
22 Feb 2015, 06:33
Smgs wrote: Bunuel wrote: kwhitejr wrote: Can anyone demonstrate the following?
If n is a positive integer and r is the remainder when (n1)(n+1) is divided by 24, what is the value of r?
(1) 2 is not a factor of n (2) 3 is not a factor of n Hi, and welcome to Gmat Club! Below is the solution for your problem: If n is a positive integer and r is the remainder when (n1)(n+1) is divided by 24, what is the value of r?Number plugging method:\((n1)(n+1)=n^21\) (1) n is not divisible by 2 > pick two odd numbers: let's say 1 and 3 > if \(n=1\), then \(n^21=0\) and as zero is divisible by 24 (zero is divisible by any integer except zero itself) so remainder is 0 but if \(n=3\), then \(n^21=8\) and 8 divided by 24 yields remainder of 8. Two different answers, hence not sufficient. (2) n is not divisible by 3 > pick two numbers which are not divisible by 3: let's say 1 and 2 > if \(n=1\), then \(n^21=0\), so remainder is 0 but if \(n=2\), then \(n^21=3\) and 3 divided by 24 yields remainder of 3. Two different answers, hence not sufficient. (1)+(2) Let's check for several numbers which are not divisible by 2 or 3: \(n=1\) > \(n^21=0\) > remainder 0; \(n=5\) > \(n^21=24\) > remainder 0; \(n=7\) > \(n^21=48\) > remainder 0; \(n=11\) > \(n^21=120\) > remainder 0. Well it seems that all appropriate numbers will give remainder of 0. Sufficient. Algebraic approach:(1) n is not divisible by 2. Insufficient on its own, but this statement says that \(n=odd\) > \(n1\) and \(n+1\) are consecutive even integers > \((n1)(n+1)\) must be divisible by 8 (as both multiples are even and one of them will be divisible by 4. From consecutive even integers one is divisible by 4: (2, 4); (4, 6); (6, 8); (8, 10); (10, 12), ...). (2) n is not divisible by 3. Insufficient on its own, but form this statement either \(n1\) or \(n+1\) must be divisible by 3 (as \(n1\), \(n\), and \(n+1\) are consecutive integers, so one of them must be divisible by 3, we are told that it's not \(n\), hence either \(n1\) or \(n+1\)). (1)+(2) From (1) \((n1)(n+1)\) is divisible by 8, from (2) it's also divisible by 3, therefore it must be divisible by \(8*3=24\), which means that remainder upon division \((n1)(n+1)\) by 24 will be 0. Sufficient. Answer: C. Hope it helps. Dear Bunuel, Thanks for sharing the alternatives to approaching this problem. I have a question here. What if we choose a number, n, such as 23 (which is neither divisible by 2 nor by 3)? In that case the given expression (n1)*(n+1) will be 22*24, which when divided by 24 will leave a remainder 22. Kindly advise! Thanks! 22* 24 is a multiple of 24, so when divided by 24 it yields the remainder of 0.
_________________
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



GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16030

Re: If n is a positive integer and r is the remainder when (n1) [#permalink]
Show Tags
03 May 2016, 20:26
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



GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16030

Re: If n is a positive integer and r is the remainder when (n1) [#permalink]
Show Tags
27 Jun 2017, 11:39
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




Re: If n is a positive integer and r is the remainder when (n1)
[#permalink]
27 Jun 2017, 11:39







