Author 
Message 
TAGS:

Hide Tags

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7934
Location: Pune, India

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
08 Oct 2013, 20:25
khairilthegreat wrote: Dear Bunuel,
"Now, to guarantee that at least one multiple is divisible by 3, these numbers must have different remainders upon division by 3"
Why is this? Why the numbers must have different reminders? Any positive number will take one of three forms: 3m, (3m+1) or (3m+2) i.e. either it will be divisible by 3, will leave remainder 1 or will leave remainder 2 when divided by 3. If the number takes the form 3m, the number after it is of the form 3m+1 and the one after it is of the form 3m+2. If we have 3 consecutive numbers such as a, (a+1), (a+2), we know for sure that at least one of them is divisible by 3 since one of them will be of the form 3m. We don't know which one but one of them will be divisible by 3. So given numbers such as (n1)*n*(n+1), we know that the product is divisible by 3. In the given options, we don't know whether n is divisible by 3 or not. We need to look for the option which has 3 consecutive numbers i.e. in which the terms leave a remainder of 0, 1 and 2 to be able to say that the product will be divisible by 3. Note a product such as n(n+3)(n+6). When this is divided by 3, we cannot say whether it is divisible or not because all three factors will leave the same remainder, 1. Say n = 4. Product 4*7*10. All these factors are of the form 3m+1. We don't have any 3m factor here. So we need the factors to have 3 different remainders so that one of them is of the form 3m.
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Manager
Joined: 07 Apr 2012
Posts: 121
Location: United States
Concentration: Entrepreneurship, Operations
GPA: 3.9
WE: Operations (Manufacturing)

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
09 Oct 2013, 04:45
(n4)+3=n1
How to conclude on this ?



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7934
Location: Pune, India

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
09 Oct 2013, 20:43
ygdrasil24 wrote: (n4)+3=n1
How to conclude on this ? When talking about 'divisibility by 3', (n  4) and (n  1) are of the same form. This means that if (n1) is divisible by 3, so is (n  4). If (n1) leaves a remainder of 1 when divided by 3, so does (n4). If (n1) leaves a remainder of 2 when divided by 3, so does (n4). the reason for this is that (n1) and (n4) have a difference of 3 between them. If (n4) = 5, (n1) = 8. Both leave remainder 2 when divided by 3 If (n4) = 6, (n1) = 9. Both leave remainder 0 when divided by 3 If (n4) = 7, (n1) = 10. Both leave remainder 1 when divided by 3
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Manager
Joined: 07 Apr 2012
Posts: 121
Location: United States
Concentration: Entrepreneurship, Operations
GPA: 3.9
WE: Operations (Manufacturing)

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
10 Oct 2013, 00:44
VeritasPrepKarishma wrote: ygdrasil24 wrote: (n4)+3=n1
How to conclude on this ? When talking about 'divisibility by 3', (n  4) and (n  1) are of the same form. This means that if (n1) is divisible by 3, so is (n  4). If (n1) leaves a remainder of 1 when divided by 3, so does (n4). If (n1) leaves a remainder of 2 when divided by 3, so does (n4). the reason for this is that (n1) and (n4) have a difference of 3 between them. If (n4) = 5, (n1) = 8. Both leave remainder 2 when divided by 3 If (n4) = 6, (n1) = 9. Both leave remainder 0 when divided by 3 If (n4) = 7, (n1) = 10. Both leave remainder 1 when divided by 3 Okay. Thanks. Its a bit awkward to get the logic initially. usually when asked for division by 3, i would look for a 3N type number. If the question had , say, to check for 4, then in that case we should be having 4 multipliers of n something like N(N2)(N+3)(N4) and so on ? So that we can have remainders as 0,1,2,3



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7934
Location: Pune, India

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
10 Oct 2013, 03:35
ygdrasil24 wrote: VeritasPrepKarishma wrote: ygdrasil24 wrote: (n4)+3=n1
How to conclude on this ? When talking about 'divisibility by 3', (n  4) and (n  1) are of the same form. This means that if (n1) is divisible by 3, so is (n  4). If (n1) leaves a remainder of 1 when divided by 3, so does (n4). If (n1) leaves a remainder of 2 when divided by 3, so does (n4). the reason for this is that (n1) and (n4) have a difference of 3 between them. If (n4) = 5, (n1) = 8. Both leave remainder 2 when divided by 3 If (n4) = 6, (n1) = 9. Both leave remainder 0 when divided by 3 If (n4) = 7, (n1) = 10. Both leave remainder 1 when divided by 3 Okay. Thanks. Its a bit awkward to get the logic initially. usually when asked for division by 3, i would look for a 3N type number. If the question had , say, to check for 4, then in that case we should be having 4 multipliers of n something like N(N2)(N+3)(N4) and so on ? So that we can have remainders as 0,1,2,3 When considering division by 4, N and N4 are the same thing. We would be looking for a product such as N(N+1)(N+2)(N+3) or (N1)N(N+1)(N2) (since N1 is the same as N+3 and N2 is the same as N+2 so this product is same as the first product) etc Check out: http://www.veritasprep.com/blog/2011/09 ... cormath/http://www.veritasprep.com/blog/2011/09 ... hpartii/The logic is explained in detail in these 2 posts.
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Manager
Joined: 07 Apr 2012
Posts: 121
Location: United States
Concentration: Entrepreneurship, Operations
GPA: 3.9
WE: Operations (Manufacturing)

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
10 Oct 2013, 04:33
OK Thanks , Can I subscribe to get updates from your blog?



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7934
Location: Pune, India

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
10 Oct 2013, 20:46
ygdrasil24 wrote: OK Thanks , Can I subscribe to get updates from your blog? Sure. You can subscribe to our blog through Twitter/Facebook/Email/News Readers http://www.veritasprep.com/blog/Check out these options in the 'Connect With Veritas Prep' box on this page
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Intern
Joined: 04 Aug 2013
Posts: 6
Concentration: Sustainability, Entrepreneurship
WE: Architecture (Energy and Utilities)

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
29 Oct 2013, 15:50
Try with 1, and that will limit us to option A & D. Alternatively, looking at all the options, only A will result to 3. This is how I did 1. N+1 + N 4 = 2n3 (Any no. will give a multiple of 3). 2. N1 + N+2 = 2n+1 3. N5 + N+3 = 2n2 4. N2 + N+4 = 2n +2 5. N6 + N+5 = 2n 1
*not sure if this is correct method.



Intern
Joined: 28 Sep 2013
Posts: 1

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
30 Oct 2013, 17:50
multiplication of three consecutive numbers is always divisible by 3, i.e (n1)*n*(n+1) or n*(n+1)*(n+2) or (n2)*(n1)*n check the option that represents any of the above types.
a) satisfies (n1)*n*(n+1) condition (n4)= (n1) because difference of both is 3 for verification substitute n=13, 14, 19, 10



Intern
Joined: 02 Jan 2014
Posts: 12

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
09 Mar 2014, 10:00
Multiple brilliant answers in this wonderful forum... as always. However... I found a way that I personally understood better, and a way I think I can use in a timeconstrained manner in the actual exams.
According to GMAT materials, the ways where an integer can be divided by 3 are 1) when the sum of the integer DIGITS is divisible by 3 2) the integer have multiple of 3's 3) a consecutive set of 3 integers can be divisible by both 2 and 3 (factor foundation rule)
There are two ways to solve this 1) Plugging in: Undoubtedly, plugging in numbers to the 5 options below is perhaps the simplest and easiest way to answer this question. To save time on computation, using the sum of integer DIGITS may accelerate your process, or looking for multiples of 3s. For example: Option A: n(n + 1)(n  4) = 7(7 +1)(7  4) = (7)(8)(3)
2) Look for patterns Given the answer options, which are adjusting the position of n on the number lines, another way is to scan if any of these answer choices provide clues to consecutive numbers.
In Option (A): we have a partial consecutive set of N, N+1. It looks like it is missing a N+2 or N1. OR is it? Now look closer at (N4). From a consecutive set perspective, (N4) is also the same "position" as (N1) Therefore this answer choice has N, N+1 and N1. This is a consecutive set that is both divisible by 2 and 3.
All other options will not have such a relationship: Option (B) n(n + 2)(n – 1): We have N, N+2, and (N1 or N+3....). Therefore missing N+1 to make a consecutive set
Option (C) n(n + 3)(n – 5) We have N, N+3, and (N5 or N2 or N+1....) Therefore missing N+2 to make a consecutive set
Option (D) n(n + 4)(n – 2) We have N, N+4 and (N2, N+1, N+4...). Therefore missing N+2 to make a consecutive set
Option (E) n(n + 5)(n – 6) We have N, N+5 and (N6, N3, N, N+3....). Therefore missing N+2 to make a consecutive set



Intern
Joined: 02 Jan 2014
Posts: 12

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
09 Mar 2014, 10:02
Multiple brilliant answers in this wonderful forum... as always. However... I found a way that I personally understood better, and a way I think I can use in a timeconstrained manner in the actual exams.
Concept According to GMAT materials, the ways where an integer can be divided by 3 are 1) when the sum of the integer DIGITS is divisible by 3 2) the integer have multiple of 3's 3) a consecutive set of 3 integers can be divisible by both 2 and 3 (factor foundation rule)
There are two ways to solve this 1) Plugging in: Undoubtedly, plugging in numbers to the 5 options below is perhaps the simplest and easiest way to answer this question. To save time on computation, using the sum of integer DIGITS may accelerate your process, or looking for multiples of 3s. For example: Option A: n(n + 1)(n  4) = 7(7 +1)(7  4) = (7)(8)(3)
2) Look for patterns Given the answer options, which are adjusting the position of n on the number lines, another way is to scan if any of these answer choices provide clues to consecutive numbers.
In Option (A): we have a partial consecutive set of N, N+1. It looks like it is missing a N+2 or N1. OR is it? Now look closer at (N4). From a consecutive set perspective, (N4) is also the same "position" as (N1) Therefore this answer choice has N, N+1 and N1. This is a consecutive set that is both divisible by 2 and 3.
All other options will not have such a relationship: Option (B): n(n + 2)(n – 1): We have N, N+2, and (N1 or N+3....). Therefore missing N+1 to make a consecutive set
Option (C): n(n + 3)(n – 5) We have N, N+3, and (N5 or N2 or N+1....) Therefore missing N+2 to make a consecutive set
Option (D): n(n + 4)(n – 2) We have N, N+4 and (N2, N+1, N+4...). Therefore missing N+2 to make a consecutive set
Option (E): n(n + 5)(n – 6) We have N, N+5 and (N6, N3, N, N+3....). Therefore missing N+2 to make a consecutive set



Retired Moderator
Joined: 29 Oct 2013
Posts: 283
Concentration: Finance
GPA: 3.7
WE: Corporate Finance (Retail Banking)

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
01 Dec 2015, 02:28
EMPOWERgmatRichC: can you also provide some insights into this question? What principle is GMAT testing here? What's the most efficient way to tackle it etc. Thanks
_________________
Please contact me for super inexpensive quality private tutoring
My journey V46 and 750 > http://gmatclub.com/forum/myjourneyto46onverbal750overall171722.html#p1367876



EMPOWERgmat Instructor
Status: GMAT Assassin/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 11019
Location: United States (CA)
GRE 1: 340 Q170 V170

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
01 Dec 2015, 20:43
Hi MensaNumber, This question can be solved rather easily by TESTing VALUES, although the work itself will take a bit longer than average and it would help a great deal if you could spot the subtle Number Properties involved. From the question stem, you can see that we're dealing with division by 3 (or the 'rule of 3', if you learned the concept that way). You don't actually have to multiply out any of the answer choices though  you just need to find the one answer that will ALWAYS have a '3' in one of its 'pieces.' The subtle Number Property I referred to at the beginning is the 'spacing out' of the terms. (1)(2)(3) is a multiple of 3, since it's 3 times some other integers. (5)(6)(7) is also a multiple of 3, since we can find a 3 'inside' the 6, so we have 3x2 times some other integers. Looking at the answer choices to this question, we're clearly NOT dealing with consecutive integers, but the 'cycle' of integers is something that we can still take advantage of. For example, we know that... When n is an integer, (n+1)(n+2)(n+3) will include a multiple of 3, since it's 3 consecutive integers (one of those 3 terms MUST be a multiple of 3, even if you don't know exactly which one it is). You can take this same concept and 'move around' any (or all) of the pieces: (n+1)(n+2)(n+6) will also include a multiple of 3 (that third term is 3 'more' than 'n+3'). Instead of adding a multiple of 3 to a term, you could also subtract a multiple of 3 from a term. eg. (n2)(n+2)(n+3) will also include a multiple of 3 (that first timer is 3 'less' than 'n+1'). The correct answer to this question subtracts a multiple of 3 from one of the terms. Final Answer: All things being equal, I'd still stick to TESTing VALUES (and not approaching the prompt with math theory)  the math is easy and you can put it 'on the pad' with very little effort. GMAT assassins aren't born, they're made, Rich
_________________
760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com
Rich Cohen
CoFounder & GMAT Assassin
Special Offer: Save $75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/
***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************



Retired Moderator
Joined: 29 Oct 2013
Posts: 283
Concentration: Finance
GPA: 3.7
WE: Corporate Finance (Retail Banking)

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
01 Dec 2015, 22:09
Thanks for your reply EMPOWERgmatRichC! I have read every single expert and non expert reply on this question on this and other forums. This by far the best and most lucid explanation I have seen. I have learnt a very powerful technique here to test for the divisibility of 3, you should see if you're able to create three consecutive numbers by arithmetic manipulation using 3. I guess similarly for divisibility by 4 we can create 4 consec numbers by arithmetic manupulation using 4 etc. To appreciate beauty of your approach I needed the theory explained so brilliantly by VeritasPrepKarishma in her blog posts on this topics. Thanks guys! I also agree that number testing could be far straightforward approach. but I felt it was time consuming so was looking for a more intuitive conceptual approach. Thanks again
_________________
Please contact me for super inexpensive quality private tutoring
My journey V46 and 750 > http://gmatclub.com/forum/myjourneyto46onverbal750overall171722.html#p1367876



Intern
Joined: 25 Feb 2017
Posts: 48
GPA: 3.67

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
01 May 2017, 16:20
If n is an integer greater than 6, which of the following must be divisible by 3 ?
(A) n(n + 1)(n – 4) (B) n(n + 2)(n – 1) (C) n(n + 3)(n – 5) (D) n(n + 4)(n – 2) (E) n(n + 5)(n – 6)
My 2 cents. Plugging in method used, total time to solve (1 min)
First, 7. a) 7,8,3 > good as there is 3 b) 7,9,8 > good c) 7,10,2 > no good d) 7,11,5 > no good e) 7,12, 1 > good
So we are left with a), b) and e)
Now, try 8 a) 8,9,4 > good b) 8,10,7 > no good e) 8,13,2 > no good
Hence, A is correct.



Manager
Joined: 12 Jun 2016
Posts: 225
Location: India
Concentration: Technology, Leadership
WE: Sales (Telecommunications)

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
28 Jun 2017, 02:11
Hello carcass! I stumbled upon your explanation while working on this OG 16 question. I found it to be quite intuitive and helpful. Sorry, for asking question on something posted so long back I have a question on this approach (relevant part highlighted). Is it necessary that if a number is divisible 3, the sum of individual digits of its factors should be divisible by 3? Lets take the case of 30 which is divisible by 30. If I write 30 = 2*5*3  the sum of digits = 2+5+3 = 10 is not divisible by 3. What am I missing here? Thanking you in advance! carcass wrote: I attacked the problem in this was, tell me if pron of errors
n is an integers, so we can choose 7, 8, 9 and so on. Now, a number divisible by 3 the sum of number MUST be divisible by 3.
1) n (n+1)(n4) > 8 * 9 * 4 > without perform multiplication the SUM of 8 + 9 + 4 = 21 and is divisible by 3 without reminder
The rest of choices do not work if you try.
thanks
_________________
My Best is yet to come!



Director
Joined: 17 Dec 2012
Posts: 627
Location: India

If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
28 Jun 2017, 03:33
carcass wrote: If n is an integer greater than 6, which of the following must be divisible by 3 ?
(A) n(n + 1)(n – 4) (B) n(n + 2)(n – 1) (C) n(n + 3)(n – 5) (D) n(n + 4)(n – 2) (E) n(n + 5)(n – 6) To solve semi logically, take n in each. In choice A, either adding 1 to n or subtracting 4 from n will give you a multiple of 3. If neither n will be a multiple of 3.So it has to be divisible by 3. To illustrate take the extreme values. In A, (n4) and (n+1) are the extreme values. There is a difference of 5. (3,8), (4,9), (5,10) are examples . (n4), (n+1) and n resp. are the multiples of 3 in these examples, making the expression divisible by 3. Seeing 2 values at a time reduces the clutter.
_________________
Srinivasan Vaidyaraman Sravna http://www.sravnatestprep.com
Premium Material Standardized Approaches
Last edited by SravnaTestPrep on 28 Jun 2017, 04:09, edited 1 time in total.



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7934
Location: Pune, India

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
28 Jun 2017, 03:46
susheelh wrote: Hello carcass! I stumbled upon your explanation while working on this OG 16 question. I found it to be quite intuitive and helpful. Sorry, for asking question on something posted so long back I have a question on this approach (relevant part highlighted). Is it necessary that if a number is divisible 3, the sum of individual digits of its factors should be divisible by 3? Lets take the case of 30 which is divisible by 30. If I write 30 = 2*5*3  the sum of digits = 2+5+3 = 10 is not divisible by 3. What am I missing here? Thanking you in advance! carcass wrote: I attacked the problem in this was, tell me if pron of errors
n is an integers, so we can choose 7, 8, 9 and so on. Now, a number divisible by 3 the sum of number MUST be divisible by 3.
1) n (n+1)(n4) > 8 * 9 * 4 > without perform multiplication the SUM of 8 + 9 + 4 = 21 and is divisible by 3 without reminder
The rest of choices do not work if you try.
thanks This is not true. 8*9*4 is divisible by 3 because one of the factors is 9. So of course the product has 3 as a factor. For more on this, check: https://www.veritasprep.com/blog/2014/0 ... rfactors/
_________________
Karishma Veritas Prep  GMAT Instructor My Blog
Get started with Veritas Prep GMAT On Demand for $199
Veritas Prep Reviews



Manager
Joined: 12 Jun 2016
Posts: 225
Location: India
Concentration: Technology, Leadership
WE: Sales (Telecommunications)

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
28 Jun 2017, 04:55
Thanks a ton for the reply VeritasPrepKarishma!! I am clear about it now. Thanks also for the GREAT blog that you maintain. I have learnt a lot from it. I am sure, I will continue to learn even more from it in future. VeritasPrepKarishma wrote: susheelh wrote: Hello carcass! I stumbled upon your explanation while working on this OG 16 question. I found it to be quite intuitive and helpful. Sorry, for asking question on something posted so long back I have a question on this approach (relevant part highlighted). Is it necessary that if a number is divisible 3, the sum of individual digits of its factors should be divisible by 3? Lets take the case of 30 which is divisible by 30. If I write 30 = 2*5*3  the sum of digits = 2+5+3 = 10 is not divisible by 3. What am I missing here? Thanking you in advance! carcass wrote: I attacked the problem in this was, tell me if pron of errors
n is an integers, so we can choose 7, 8, 9 and so on. Now, a number divisible by 3 the sum of number MUST be divisible by 3.
1) n (n+1)(n4) > 8 * 9 * 4 > without perform multiplication the SUM of 8 + 9 + 4 = 21 and is divisible by 3 without reminder
The rest of choices do not work if you try.
thanks This is not true. 8*9*4 is divisible by 3 because one of the factors is 9. So of course the product has 3 as a factor. For more on this, check: https://www.veritasprep.com/blog/2014/0 ... rfactors/
_________________
My Best is yet to come!



Manager
Joined: 09 Mar 2016
Posts: 212

Re: If n is an integer greater than 6, which of the following mu [#permalink]
Show Tags
25 Nov 2017, 05:03
Bunuel wrote: carcass wrote: If n is an integer greater than 6, which of the following must be divisible by 3 ?
(A) n(n + 1)(n – 4) (B) n(n + 2)(n – 1) (C) n(n + 3)(n – 5) (D) n(n + 4)(n – 2) (E) n(n + 5)(n – 6) Since 3 is a prime number then in order the product to be divisible by 3 either of the multiples must be divisible by 3. Now, to guarantee that at least one multiple is divisible by 3, these numbers must have different remainders upon division by 3, meaning that one of them should have the remainder of 1, another the reminder of 2 and the third one the remainder of 0, so be divisible by 3. For option A: n and n+1 have different remainder upon division by 3. As for n4, it will have the same remainder as (n4)+3=n1, so also different than the remainders of the previous two numbers. Answer: A. Similar question to practice: http://gmatclub.com/forum/ifxisanin ... 26853.htmlHope it helps. Bunuel good day! can you please explain this part "it will have the same remainder as (n4)+3=n1, so also different than the remainders of the previous two numbers" I dont understand the logic behind this (n4)+3=n1 have a great day thanks!




Re: If n is an integer greater than 6, which of the following mu
[#permalink]
25 Nov 2017, 05:03



Go to page
Previous
1 2 3
Next
[ 43 posts ]



