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# If n and k are positive integers, is n divisible by 6?

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If n and k are positive integers, is n divisible by 6? [#permalink]

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27 Aug 2009, 17:15
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If n and k are positive integers, is n divisible by 6?

(1) n = k(k + 1)(k - 1)

(2) k – 1 is a multiple of 3
[Reveal] Spoiler: OA
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Re: Is n divisible by 6? [#permalink]

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27 Aug 2009, 17:22
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1.) Sufficient

Always remember, when 3 consecutive integers are multiplied, then one should be a multiple of 3 and atleast one (or max 2) number is even. Also, if the multiple of 3 is also the even number like 6, 12, 18 then it is divisible by 6 always..

For ex:- 1*2*3 ; -4*-5*-6 ; 4*5*6 ; 11*12*13..

In any case; the product would be divisible by 6.

2.) insufficient.. does not say anything about n..

Ans. A
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Re: Is n divisible by 6? [#permalink]

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27 Aug 2009, 21:47
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tarek99 wrote:
If n and k are positive integers, is n divisible by 6?

(1) n = k(k + 1)(k - 1)
(2) k – 1 is a multiple of 3

Thanks

My answer would be A.
Since n = product of 3 consecutive numbers, it will be divisible by 3. We should also remember that 0 is also a multiple of 6. For eg : if k = 1,
n = 1*2*0 => n = 0 which IS divisible by 6.

I hope I am right. "0 is divisible by any number" right?
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Re: Is n divisible by 6? [#permalink]

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27 Aug 2009, 23:50
(1) n = k(k + 1)(k - 1) --- Sufficient
(2) k – 1 is a multiple of 3 --- insufficient
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Re: Is n divisible by 6? [#permalink]

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28 Aug 2009, 03:30
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bhanushalinikhil wrote:
tarek99 wrote:
If n and k are positive integers, is n divisible by 6?

(1) n = k(k + 1)(k - 1)
(2) k – 1 is a multiple of 3

Thanks

My answer would be A.
Since n = product of 3 consecutive numbers, it will be divisible by 3. We should also remember that 0 is also a multiple of 6. For eg : if k = 1,
n = 1*2*0 => n = 0 which IS divisible by 6.

I hope I am right. "0 is divisible by any number" right?

I slightly differ on your explination......specially if k = 1...

If k = 1 then n = 0*1*2 then n = 0 ... zero is not a positive integer hence all values of k will start from 2 onwards
hence values of n will start from 6 onwards.....but yes your ans is right A it is
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Re: Is n divisible by 6? [#permalink]

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28 Aug 2009, 04:35
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bhushan252 wrote:
bhanushalinikhil wrote:
tarek99 wrote:
If n and k are positive integers, is n divisible by 6?

(1) n = k(k + 1)(k - 1)
(2) k – 1 is a multiple of 3

Thanks

My answer would be A.
Since n = product of 3 consecutive numbers, it will be divisible by 3. We should also remember that 0 is also a multiple of 6. For eg : if k = 1,
n = 1*2*0 => n = 0 which IS divisible by 6.

I hope I am right. "0 is divisible by any number" right?

I slightly differ on your explination......specially if k = 1...

If k = 1 then n = 0*1*2 then n = 0 ... zero is not a positive integer hence all values of k will start from 2 onwards
hence values of n will start from 6 onwards.....but yes your ans is right A it is

Right. Now see the mistake.
Thanks!
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Re: Is n divisible by 6? [#permalink]

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15 Apr 2014, 07:54
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Re: If n and k are positive integers, is n divisible by 6? [#permalink]

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24 Apr 2015, 16:17
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Re: If n and k are positive integers, is n divisible by 6? [#permalink]

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16 May 2015, 04:15
Unsure how A is sufficient (I know that's the answer but I'm hoping someone could help show me why)

For me A is insufficient because:

if n= k(k+1)(k-1)

we can get several cases, such as

k= 0
0*1*(-1) = -1 <-- not divisible by 6

k=1
1*2*0 = 0 <-- not divisible by 6

k=2
2*3*1 = 6 <--- divisible by 6

And so on. Statement 2 could supplement to my dilemma here, but since the answer is A, does anyone have a dumbed down explanation?
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Re: If n and k are positive integers, is n divisible by 6? [#permalink]

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16 May 2015, 04:22
sabineodf wrote:
Unsure how A is sufficient (I know that's the answer but I'm hoping someone could help show me why)

For me A is insufficient because:

if n= k(k+1)(k-1)

we can get several cases, such as

k= 0
0*1*(-1) = -1 <-- not divisible by 6

k=1
1*2*0 = 0 <-- not divisible by 6

k=2
2*3*1 = 6 <--- divisible by 6

And so on. Statement 2 could supplement to my dilemma here, but since the answer is A, does anyone have a dumbed down explanation?

JUST realised that the two first instances that I listed up are NOT positive integers hence they cannot be N. Ooops. Forget my question
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Re: If n and k are positive integers, is n divisible by 6? [#permalink]

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16 May 2015, 04:54
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sabineodf wrote:
Unsure how A is sufficient (I know that's the answer but I'm hoping someone could help show me why)

For me A is insufficient because:

if n= k(k+1)(k-1)

we can get several cases, such as

k= 0
0*1*(-1) = -1 <-- not divisible by 6 = 0

k=1
1*2*0 = 0 <-- not divisible by 6

k=2
2*3*1 = 6 <--- divisible by 6

And so on. Statement 2 could supplement to my dilemma here, but since the answer is A, does anyone have a dumbed down explanation?

0 is divisible by every integer (except 0 itself).
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Re: If n and k are positive integers, is n divisible by 6? [#permalink]

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16 May 2015, 06:18
Bunuel wrote:
sabineodf wrote:
Unsure how A is sufficient (I know that's the answer but I'm hoping someone could help show me why)

For me A is insufficient because:

if n= k(k+1)(k-1)

we can get several cases, such as

k= 0
0*1*(-1) = -1 <-- not divisible by 6 = 0

k=1
1*2*0 = 0 <-- not divisible by 6

k=2
2*3*1 = 6 <--- divisible by 6

And so on. Statement 2 could supplement to my dilemma here, but since the answer is A, does anyone have a dumbed down explanation?

0 is divisible by every integer (except 0 itself).

Oh! Thanks, that´s good to know :D
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Re: If n and k are positive integers, is n divisible by 6? [#permalink]

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23 May 2015, 02:54
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tarek99 wrote:
If n and k are positive integers, is n divisible by 6?

(1) n = k(k + 1)(k - 1)

(2) k – 1 is a multiple of 3

Here's my way to solve this:

Statement (1):
Rewrite n = k(k + 1)(k - 1) to: $$k^3-k$$ and plug in various positive integers to see that the result will alsways be divisible by 3. Therefore AC 1 = Sufficient

Statement (2)
Gives you just an idea about the term k-1, but nothing about k or k+1 itself. Therefore clearly insufficient.
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If n and k are positive integers, is n divisible by 6? [#permalink]

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19 Sep 2015, 05:17
i dont understand why b is not sufficient,
if k-1 is a multiple of 3, then k or k+1 should be a multiple of 2...
and so, it should be sufficient as well....
am i missing something?
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Re: If n and k are positive integers, is n divisible by 6? [#permalink]

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19 Sep 2015, 05:53
saroshgilani wrote:
i dont understand why b is not sufficient,
if k-1 is a multiple of 3, then k or k+1 should be a multiple of 2...
and so, it should be sufficient as well....
am i missing something?

You are correct to say that if k-1 =3p, then k or k+1 will be =2r. But what is the connection between k and n? You are using information from statement 1 to check for sufficiency of statement 2. This absolutely wrong. You need to isolate both statements completely to determine their respective sufficiency. Only once they are not sufficient ALONE, you go to combining the 2 statements.
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If n and k are positive integers, is n divisible by 6? [#permalink]

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19 Sep 2015, 06:01
reto wrote:
tarek99 wrote:
If n and k are positive integers, is n divisible by 6?

(1) n = k(k + 1)(k - 1)

(2) k – 1 is a multiple of 3

Here's my way to solve this:

Statement (1):
Rewrite n = k(k + 1)(k - 1) to: $$k^3-k$$ and plug in various positive integers to see that the result will alsways be divisible by 3. Therefore AC 1 = Sufficient

Statement (2)
Gives you just an idea about the term k-1, but nothing about k or k+1 itself. Therefore clearly insufficient.

reto, a couple of points about your solution (although you were able to get to the correct answer):

For statement 1, k(k-1)(k+1) will ALWAYS be divisible by 6. Think of this way. For any number to be divisible by 6, it needs to be divisible by BOTH 2 and 3 at the same time.

Consider 2 cases, if k =2p, then k+1 or k-1 MUST be disivible by 3. Thus the product k(k-1)(k+1) will always be divisible by 6.

Case 2: if k = 3p, then k-1 or k+1 MUST be divisble by 2. Thus the product k(k-1)(k+1) will always be divisible by 6.

As for your interpretation of statement 2, we do not need to even look at the statement and start computing numbers as the question stem asks us whether "n" is divisible by 6 and NOT whether "k" is divisible by 6. Statement 2 does not provide any relation between k and n and hence straightaway you can reject this statement.

Also you are incorrect to say that as k-1=3p, you have no information about k+1 or k. This is not correct. If k-1=3p., then 1 of k or k+1 MUST be divisible by 2 (you can check with some numbers). Thus you do get some actionable information (albeit not pertinent to the original question asked!).
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Re: If n and k are positive integers, is n divisible by 6? [#permalink]

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19 Sep 2015, 06:37
Engr2012 wrote:
Also you are incorrect to say that as k-1=3p, you have no information about k+1 or k. This is not correct. If k-1=3p., then 1 of k or k+1 MUST be divisible by 2 (you can check with some numbers). Thus you do get some actionable information (albeit not pertinent to the original question asked!).

Hi Engr2012,
irrespective of what K-1 is, one of k or k+1 will always be divisible by 2 as k is a positive integer..
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If n and k are positive integers, is n divisible by 6? [#permalink]

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19 Sep 2015, 06:38
chetan2u wrote:
Engr2012 wrote:
Also you are incorrect to say that as k-1=3p, you have no information about k+1 or k. This is not correct. If k-1=3p., then 1 of k or k+1 MUST be divisible by 2 (you can check with some numbers). Thus you do get some actionable information (albeit not pertinent to the original question asked!).

Hi Engr2012,
irrespective of what K-1 is, one of k or k+1 will always be divisible by 2 as k is a positive integer..

Yes that is what I have written above. Additionally, the question asks about n and NOT k. Without any relation between n and k , this information is of no use.
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Re: If n and k are positive integers, is n divisible by 6? [#permalink]

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19 Sep 2015, 06:43
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chetan2u wrote:
Engr2012 wrote:
Also you are incorrect to say that as k-1=3p, you have no information about k+1 or k. This is not correct. If k-1=3p., then 1 of k or k+1 MUST be divisible by 2 (you can check with some numbers). Thus you do get some actionable information (albeit not pertinent to the original question asked!).

Hi Engr2012,
irrespective of what K-1 is, one of k or k+1 will always be divisible by 2 as k is a positive integer..

Yes but which one will it be ? Additionally, the question asks about n and NOT k. Without any relation between n and k , this information is of no use.

Hi,
I am not questioning the solution. it is very clear and fairly simple...
i am just pointing to the highlighted portion that it is not because k-1=3p that k or k+1 will be div by 2 but one of k or k+1 will always be div by 2...
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Re: If n and k are positive integers, is n divisible by 6? [#permalink]

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09 Nov 2015, 08:21
Bunuel wrote:

0 is divisible by every integer (except 0 itself).

Hi Bunuel

I just realized something while solving this question, please let me know if im correct on this one:

Product of n consecutive integers is always divisible by
-all integers between 1 and n inclusive.
-n!
-all the factors of n!

Can we make this generalization??
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Re: If n and k are positive integers, is n divisible by 6?   [#permalink] 09 Nov 2015, 08:21

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