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# If n is an integer greater than 6, which of the following mu

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If n is an integer greater than 6, which of the following mu [#permalink]

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20 Sep 2012, 17:26
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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)
[Reveal] Spoiler: OA

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Last edited by Bunuel on 21 Sep 2012, 01:13, edited 1 time in total.
Edited the question.
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Re: If n is an integer greater than 6, which of the following mu [#permalink]

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03 Jul 2013, 21:51
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pavan2185 wrote:
Which method is better in questions like these to apply on the test? to pick the numbers or to apply concept of divisibility by 3 ( Remainder 0,1,2)

In my opinion, using logic is almost always better than plugging in. Plugging in is full of possibilities of making mistakes - incorrect calculation, not considering all possibilities, getting lost in the options etc.

Logic is far cleaner.

I know that talking about positive integers, in any set of 3 consecutive positive integers, one integer will be divisible by 3 and the other 2 will not be.

So I am looking for 3 consecutive positive integers e.g. (n-1)n(n+1)

Note that (n-4) is equivalent to (n-1) since if (n-4) is divisible by 3, so is (n-1). If (n-4) is not divisible by 3, neither is (n-1) (because the difference between these two integers is 3)
Hence (A) is equivalent to 3 consecutive integers.

On same lines, note that n is equivalent to (n-3), (n + 3), (n + 6) etc.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Moderator Joined: 01 Sep 2010 Posts: 3010 Followers: 741 Kudos [?]: 5963 [5] , given: 977 Re: If n is an integer greater than 6, which of the following mu [#permalink] ### Show Tags 21 Sep 2012, 06:54 5 This post received KUDOS 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)(n-4) ---> 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 _________________ Manager Status: Oh GMAT ! I give you one more shot :) Joined: 14 Feb 2013 Posts: 96 Location: United States (MI) Concentration: General Management, Technology GMAT 1: 580 Q44 V28 GMAT 2: 690 Q49 V34 GPA: 3.5 WE: Information Technology (Computer Software) Followers: 0 Kudos [?]: 92 [4] , given: 18 Re: If n is an integer greater than 6, which of the following mu [#permalink] ### Show Tags 31 Mar 2013, 05:55 4 This post received KUDOS 2 This post was BOOKMARKED Since this is a must be true question it must be true for any n and as we know any number is divisible by 3 if the sum of its digits is divisible by 3, so if we add up the digits or terms in the options we get the answer (A) n + (n + 1) + (n – 4) = 3n - 3 ---- divisible by 3 for any n (B) n + (n + 2) + (n – 1) = 3n + 1 (C) n + (n + 3) + (n – 5) = 3n - 2 (D) n + (n + 4) + (n – 2) = 3n + 2 (E) n + (n + 5) + (n – 6) = 3n - 1 _________________ Life is a highway I wanna ride it all night long Last edited by prasun9 on 08 Oct 2013, 05:38, edited 2 times in total. Math Expert Joined: 02 Sep 2009 Posts: 34909 Followers: 6515 Kudos [?]: 83238 [2] , given: 10146 Re: If n is an integer greater than 6, which of the following mu [#permalink] ### Show Tags 21 Sep 2012, 01:13 2 This post received KUDOS Expert's post 2 This post was BOOKMARKED 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 n-4, it will have the same remainder as (n-4)+3=n-1, so also different than the remainders of the previous two numbers. Answer: A. Similar question to practice: if-x-is-an-integer-then-x-x-1-x-k-must-be-evenly-divisible-126853.html Hope it helps. _________________ GMAT Pill Representative Joined: 14 Apr 2009 Posts: 2018 Location: New York, NY Followers: 379 Kudos [?]: 1230 [2] , given: 8 Re: If n is an integer greater than 6, which of the following mu [#permalink] ### Show Tags 24 Sep 2012, 11:39 2 This post received KUDOS Expert's post Shortcut: In every set of 3 consecutive numbers, ONE of them must be divisible by 3 when we are multiplying each of the digits 1*2*3 2*3*4 3*4*5 4*5*6 (A) n(n + 1)(n – 4) The easiest is if we have something like n(n+1)(n+2) We know in this case we DEFINITELY have an expression that is divisible by 3. n=1 => 1*2*3 n=2 => 2*3*4 n=3 => 3*4*5 All are divisible by 3. Any expression must pass our 3 consecutive integer test. n(n + 1)(n – 4) n=1 => 1*2*-3 n=2 => 2*3*1 n=3 => 3*4*-1 Even if n = 4 we have: 4*5*0 = 0 which is divisible by 3. If n = 16 16*17*12 is divisible by 3. So (A) passes all the tests and one of numbers in the expression will be divisible by 3 so the whole expression when multiplied together will be divisible by 3. _________________ Senior Manager Joined: 13 Aug 2012 Posts: 464 Concentration: Marketing, Finance GMAT 1: Q V0 GPA: 3.23 Followers: 25 Kudos [?]: 394 [2] , given: 11 Re: If n is an integer greater than 6, which of the following mu [#permalink] ### Show Tags 26 Sep 2012, 00:02 2 This post received KUDOS 1 This post was BOOKMARKED Every three consecutive positive integers will have one value divisible by 3. Attachments solution mixture.jpg [ 35.62 KiB | Viewed 13438 times ] _________________ Impossible is nothing to God. Intern Joined: 31 May 2012 Posts: 11 GMAT 1: Q45 V45 GMAT 2: Q51 V46 Followers: 0 Kudos [?]: 9 [1] , given: 8 Re: If n is an integer greater than 6, which of the following mu [#permalink] ### Show Tags 20 Sep 2012, 18:28 1 This post received KUDOS 1 This post was BOOKMARKED 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) I like this one.I do not find it discussed earlier on the board. OA later Pick odd and even numbers, e.g. 7 and 8: A) (7)(8)(3) Y ; (8)(9)(5) Y B) (7)(9)(6) Y ; (8)(10)(7) N C) (7)(10)(2) N; No need to check for 8 D) (7)(11)(5) N; No need to check for 8 E) (7)(12)(1) Y; (8)(13)(2) N Only A works for both cases. A Math Expert Joined: 02 Sep 2009 Posts: 34909 Followers: 6515 Kudos [?]: 83238 [1] , given: 10146 Re: If n is an integer greater than 6, which of the following mu [#permalink] ### Show Tags 24 Sep 2012, 04:06 1 This post received KUDOS Expert's post Jp27 wrote: Bunuel wrote: carcass wrote: If n is an integer greater than 6, which of the following must be divisible by 3 ? 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 n-4, it will have the same remainder as (n-4)+3=n-1, so also different than the remainders of the previous two numbers. Hi Bunuel - I didn't quite follow this logic. Could you please elaborate. Thanks. Cheers Can you please tell me which part didn't you understand? Meanwhile check this question: if-x-is-an-integer-then-x-x-1-x-k-must-be-evenly-divisible-126853.html It might help. _________________ Intern Joined: 15 Jan 2013 Posts: 39 Concentration: Finance, Operations GPA: 4 Followers: 0 Kudos [?]: 27 [1] , given: 6 Re: If n is an integer greater than 6, which of the following mu [#permalink] ### Show Tags 08 Feb 2013, 08:56 1 This post received KUDOS 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) The best way to tackle these kind of questions is by assuming values.. Since its says, n has to be greater than 6...assume the value of n to be 7....by this u will be able to eliminate options 3 and 4... Now take the value of n as 8....by this u will be able to eliminate options 2 and 5... Now we are left with only option 1...and that is the answer.. Senior Manager Joined: 17 Dec 2012 Posts: 442 Location: India Followers: 23 Kudos [?]: 365 [1] , given: 14 Re: If n is an integer greater than 6, which of the following mu [#permalink] ### Show Tags 03 Jul 2013, 20:58 1 This post received KUDOS We need to check only for 3 values of n because the answer will be the same for every 3 consecutive integers. Actually we need not consider one of those values as it is always a multiple of 3. So we can take the values of n as 7 and 8 or 10 and 11 or 13 and 14 etc If we plug in the values 7 and 8 we get the answer once we check A as it satisfies both the values. We need not actually check other choices as they would fail for either 7 or 8. _________________ Srinivasan Vaidyaraman Sravna http://www.sravnatestprep.com Classroom and Online Coaching Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 6925 Location: Pune, India Followers: 1997 Kudos [?]: 12435 [1] , given: 221 Re: If n is an integer greater than 6, which of the following mu [#permalink] ### Show Tags 08 Oct 2013, 21:25 1 This post received KUDOS Expert's post 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 (n-1)*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

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

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10 Oct 2013, 21:46
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ygdrasil24 wrote:
OK Thanks , Can I subscribe to get updates from your blog?

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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 7517 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: 340 Q170 V170 Followers: 332 Kudos [?]: 2223 [1] , given: 162 Re: If n is an integer greater than 6, which of the following mu [#permalink] ### Show Tags 01 Dec 2015, 21:43 1 This post received KUDOS Expert's post 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. (n-2)(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: [Reveal] Spoiler: A 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 _________________ # Rich Cohen Co-Founder & GMAT Assassin # Special Offer: Save$75 + GMAT Club Tests

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

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22 Sep 2012, 11:06
Bunuel wrote:
carcass wrote:
If n is an integer greater than 6, which of the following must be divisible by 3 ?

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 n-4, it will have the same remainder as (n-4)+3=n-1, so also different than the remainders of the previous two numbers.

Hi Bunuel - I didn't quite follow this logic. Could you please elaborate. Thanks.

Cheers
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Re: If n is an integer greater than 6, which of the following mu [#permalink]

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24 Sep 2012, 07:56
Bunuel wrote:

Can you please tell me which part didn't you understand?

Meanwhile check this question: if-x-is-an-integer-then-x-x-1-x-k-must-be-evenly-divisible-126853.html It might help.

Thanks for your response. the link really helped. Im able to solve these problems.. thanks

cheers
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Re: If n is an integer greater than 6, which of the following mu [#permalink]

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25 Sep 2012, 21:36
gmatpill wrote:
Shortcut: In every set of 3 consecutive numbers, ONE of them must be divisible by 3 when we are multiplying each of the digits

1*2*3
2*3*4
3*4*5
4*5*6

(A) n(n + 1)(n – 4)

The easiest is if we have something like n(n+1)(n+2)
We know in this case we DEFINITELY have an expression that is divisible by 3.
n=1 => 1*2*3
n=2 => 2*3*4
n=3 => 3*4*5

All are divisible by 3.

Any expression must pass our 3 consecutive integer test.

n(n + 1)(n – 4)
n=1 => 1*2*-3
n=2 => 2*3*1
n=3 => 3*4*-1

Even if n = 4 we have:
4*5*0 = 0 which is divisible by 3.

If n = 16
16*17*12 is divisible by 3.

So (A) passes all the tests and one of numbers in the expression will be divisible by 3 so the whole expression when multiplied together will be divisible by 3.

Thanks gmatpill - I have a basic question here ->

11 has a reminder 4 on division by 7
12 has a reminder 5 on division by 7
What is the reminder x * y divide by 7? - I can multiply reminders as long as I correct the excess amount... so 4 *5 = 20
20 - 7*2 = 6, which is the reminder when x * y / 7 - SO FAR CORRECT?

Reminder when 99 / 15 give me a reminder of 9
But when i factorize the den into primes = 9 * 11 / 3 * 5 => 9/3 leaves reminder 0 and 11/5 reminder leaves 1
On multiplying the reminders, we get zero. So I should not factorize the denominator unless the denominator factor into the same primes

IS THIS UNDERSTANDING CORRECT? MANY THANKS.
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Re: If n is an integer greater than 6, which of the following mu [#permalink]

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01 Oct 2012, 09:11
I tried plugging in odd and even values for n and got the answer A ...

If we assume n as being either 7 , 8 or 9 we can easily see why A is the correct answer .. Because in every instance , either of the 3 numbers multiplied will be divisible by 3 and the entire product will meet the requirements..

if n =7 , then n-4 is divisible by 3 ...
If n= 8 then n+1 is divisible by 3
If n= 9 , then n is divisible by 3 ...
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Re: If n is an integer greater than 6, which of the following mu [#permalink]

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30 Dec 2012, 10:14
mbaiseasy wrote:
Every three consecutive positive integers will have one value divisible by 3.

Beautiful solution Irene,

Picturing the number line is a 15 second approach to this problem.
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Re: If n is an integer greater than 6, which of the following mu [#permalink]

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08 Feb 2013, 07:14
just draw a table for each ans option plug in starting 7,8,9 etc. It should not take more than 75 s.
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Re: If n is an integer greater than 6, which of the following mu   [#permalink] 08 Feb 2013, 07:14

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