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# M23-12

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Math Expert
Joined: 02 Sep 2009
Posts: 51258

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16 Sep 2014, 00:18
1
9
00:00

Difficulty:

85% (hard)

Question Stats:

52% (01:26) correct 48% (01:23) wrong based on 165 sessions

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If $$n$$ is a positive integer is $$n^2-n$$ divisible by 12 ?

(1) $$n$$ is divisible by 11.

(2) $$n$$ is divisible by 19.

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Joined: 02 Sep 2009
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16 Sep 2014, 00:18
1
Official Solution:

First notice that $$n^2-n=n(n-1)$$.

(1) $$n$$ is divisible by 11. If $$n=11$$, then $$n(n-1)=11*10$$ and the answer is NO but if $$n=11*12$$, then $$n(n-1)=11*12*(11*12-1)$$ and the answer is YES. Not sufficient.

(2) $$n$$ is divisible by 19. If $$n=19$$, then $$n(n-1)=19*18$$ and the answer is NO but if $$n=19*12$$, then $$n(n-1)=19*12*(19*12-1)$$ and the answer is YES. Not sufficient.

(1)+(2) From above we have that $$n$$ is divisible by $$11*19=209$$. Now, if $$n=11*19$$, then $$n(n-1)=11*19*(209-1)=11*19*208$$ and the answer is NO (since $$11*19*208$$ is not divisible by 3, then it's not divisible by $$12=3*4$$ either) but if $$n=11*19*12$$, then $$n(n-1)=11*19*12*(11*19*12-1)$$ and the answer is YES. Not sufficient.

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24 Nov 2016, 23:48
Can you explain how you know that n(n−1)=11∗12∗(11∗12−1) is divisible by 12?
Math Expert
Joined: 02 Sep 2009
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24 Nov 2016, 23:57
rananava wrote:
Can you explain how you know that n(n−1)=11∗12∗(11∗12−1) is divisible by 12?

Because it has 12 as a multiple.
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07 May 2018, 18:24
The explanation is good. This is a high quality question.
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Kindly press the +1Kudos if you like the explanation. Thanks a lot!!!

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Joined: 11 May 2018
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05 Jul 2018, 19:18
My try is by doing with statement.
1. for being divisible by 11 let try with number 11. 11^2-11=110/12 which is not divisible. so first statement cannot be true.
2.for being divisible by 19 let try with number 19. 19^2-19=342/12 which is not divisible. so 2nd statement cannot be true.

now let combine both 11*19
for being divisible by 11*19 let try with LCM of both number 209. 209^2-209=209(209-1)/12= (209*208)/12 since none of these 208 and 209 is divisible by 3 thus t is not divisible by 12 too. so even combining these two wont solve problem.
so E is only ans left

LET ME KNOW IF I AM WRONG.
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Joined: 02 Aug 2009
Posts: 7108

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05 Jul 2018, 19:33
1
aravsha wrote:
My try is by doing with statement.
1. for being divisible by 11 let try with number 11. 11^2-11=110/12 which is not divisible. so first statement cannot be true.
2.for being divisible by 19 let try with number 19. 19^2-19=342/12 which is not divisible. so 2nd statement cannot be true.

now let combine both 11*19
for being divisible by 11*19 let try with LCM of both number 209. 209^2-209=209(209-1)/12= (209*208)/12 since none of these 208 and 209 is divisible by 3 thus t is not divisible by 12 too. so even combining these two wont solve problem.
so E is only ans left

LET ME KNOW IF I AM WRONG.

You are wrong the way you have written..
in DS, a statement is sufficient if it can give a definite answer - definite YES or Definite NO
here you have just worked on getting a NO as the answer. you have to further check if it is true that there is some case where it will be div by 12 too
if there is a case , it is insufficient then only
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: M23-12 &nbs [#permalink] 05 Jul 2018, 19:33
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# M23-12

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