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K is a set of numbers such that (i) If x is in K, then -x

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K is a set of numbers such that (i) If x is in K, then -x [#permalink] New post 07 Jul 2010, 14:48
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K is a set of numbers such that

(i) If x is in K, then -x is in K, and
(ii) if each of x and y is in K, then xy is in K

Is 12 in K?

(1) 2 is in K
(2) 3 is in K

[Reveal] Spoiler:
for (1) know that 2, -2 is in K
for (2) know that 3, -3 is in K
Together have [-3, -2, 2, 3, 6]

So I would say neither is sufficient??
[Reveal] Spoiler: OA

Last edited by Bunuel on 29 Oct 2013, 05:32, edited 2 times in total.
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Re: OG Quant #70 [#permalink] New post 07 Jul 2010, 16:07
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afyl128 wrote:
My First post :-D

K is a set of numbers such that

(i) If x is in K, then -x is in K, and
(ii) if each of x and y is in K, then xy is in K

Is 12 in K?

(1) 2 is in K
(2) 3 is in K

for (1) know that 2, -2 is in K
for (2) know that 3, -3 is in K
Together have [-3, -2, 2, 3, 6]

So I would say neither is sufficient??


Hi, and welcome to the club. Below is the solution for your problem.

(1) 2 is in K --> according to (i) -2 is n K --> according to (ii) -2*2=-4 is in K --> according to (i) -(-4)=4 is in K and so on. Thus we know that 2, -2, -4, 4, 8, -8, 16, -16, ... are in K, so basically powers of 2 and their negative pairs. Is 12 in K? We don't know. Not sufficient.

(2) 3 is in K --> according to (i) -3 is n K --> according to (ii) -3*3=-9 is in K --> according to (i) -(-9)=9 is in K and so on. Thus we know that 3, -3, -9, 9, 27, -27, 81, -81, ... are in K, so basically powers of 3 and their negative pairs. Is 12 in K? We don't know. Not sufficient.

(1)+(2) From (1) 4 is in K and from (2) 3 is in K, hence according to (ii) 4*3=12 must also be in K. Sufficient.

Answer: C.

Hope it's clear.
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Re: K is a set of numbers such that (i) If x is in K, then -x [#permalink] New post 05 Sep 2012, 02:37
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Just in case someone made my same mistake: always write down the numbers (ex. 2; -2; -4 and so on) in order to realize that you have to deal with a new number everytime you have one: if 2 is there, -2 is there; so -4 is there, so 4 is there and so on, just like Bunuel showed.

I did not write down any of that and ended up with E.

Kudos and thanks to Bunuel.
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Re: K is a set of numbers such that (i) If x is in K, then -x [#permalink] New post 17 Dec 2012, 22:44
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Sachin9 wrote:
bunuel,

This is how I solved..

For 12 to be in the set, 2 and 3 must be there.. as 12's prime factors are 2 and 3.. hence C.
Is this correct?


Responding to a pm:

Given in the question:

(ii) if each of x and y is in K, then xy is in K

What does this mean? It means that if x and y are in K, then xy must also be there e.g. x = 6, y = 8
If 6 and 8 are in K, 6*8 = 48 must also be in K.

Does it also mean that 2 and 3 (i.e. factors of 6) must also be in K? No. It is not necessary.
I am building the set K. I could have put 6 in on my own. I don't need to start from 2 and 3 necessarily. If 6 and 8 are in the set K, I necessarily need to put their product in too. But I needn't put in their prime factors. We do not know whether their prime factors were put in and hence 6 and 8 were obtained or whether they were put in by set maker's choice.

So, the statement 'if each of x and y is in K, then xy is in K' only implies that product of x and y must be in K too. It doesn't imply that factors of x and y must be in K. x and y could have been put in by choice. Who says that only prime factors can be added to the set? You can pick any number and add it to the set. The only thing is that once you put in that number, you must put in its product with every number already there in the set and so on...

As pointed out by Bunuel, I hope you see that your logic is not correct.
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Re: OG Quant #70 [#permalink] New post 07 Jul 2010, 16:47
Ahh I guess so but I'm having difficulty understanding why x * -x is in the set because it states xy is in the set and not x* -x everything else below made sense.

I'm just starting out so hopefully I get a better sense of these conditions as I progress

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Re: OG Quant #70 [#permalink] New post 07 Jul 2010, 16:57
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afyl128 wrote:
Ahh I guess so but I'm having difficulty understanding why x * -x is in the set because it states xy is in the set and not x* -x everything else below made sense.

I'm just starting out so hopefully I get a better sense of these conditions as I progress

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x and y just represent some different numbers in the set K. (ii) says that if two different numbers (x and y) are in the set, then their product (xy) is also in the set.

For example: as we know that 2 and -2 (two different numbers) are in the set, then their product (-2*2=-4) must also be in the set.

Hope it helps.
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Re: OG Quant #70 [#permalink] New post 08 Jul 2010, 14:32
many thanks for the responses!

Although I'm still missing something fundamental, if (2, -2 and -4) is in the set, why is (2 * -4 = -8 etc) in the set? As you mentioned below we know 2 numbers are in the set and the multiple of the two numbers. Is there wording in there that implies every number in the set can be multiplied by any other number in the set apart from just x or y. To me x, y and xy implies 3 values.
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Re: OG Quant #70 [#permalink] New post 08 Jul 2010, 15:57
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afyl128 wrote:
Is there wording in there that implies every number in the set can be multiplied by any other number in the set apart from just x or y.


Yes, for ANY two numbers in the set their product is also in the set.
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Re: QR DS 70 Number properties set [#permalink] New post 21 Feb 2011, 09:05
I. 2 is in k; -2,-4,4,8,-8,16,-16. 12 is not there in this series. But, it may be there.
II. 3 is in k; 3,-3,-9,9,27,-27. 12 is not there in this series. But, it may be there.

Using both; 2,3,-2,-3,-4,-9,6,-6,12. 12 is definitely there.

Ans: "C".
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Re: QR DS 70 Number properties set [#permalink] New post 12 Jun 2011, 11:50
fluke wrote:
I. 2 is in k; -2,-4,4,8,-8,16,-16. 12 is not there in this series. But, it may be there.
II. 3 is in k; 3,-3,-9,9,27,-27. 12 is not there in this series. But, it may be there.

Using both; 2,3,-2,-3,-4,-9,6,-6,12. 12 is definitely there.

Ans: "C".


Hi Fluke,

The question asks us if '12 is in K'. After writing out the sequence (as Bunnel has done), I was NOT able to get a 12. Therefore, my answer was suffcient, because I could answer definitively and say that 12 is NOT in K. I reached the same conclusion with statement (2). Therefore, my answer to this question was D.

In your quote above you mention that "[12] may be there". This is what I do not understand. If the statement given to us is a fact, then how can we assume that the sequence COULD have a 12 ? (even after we draw out the set and no 12 is present)

Bunnel, if you can, can you jump in on this one too please!
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Re: QR DS 70 Number properties set [#permalink] New post 11 Dec 2011, 02:55
ngbrian85 wrote:
fluke wrote:
I. 2 is in k; -2,-4,4,8,-8,16,-16. 12 is not there in this series. But, it may be there.
II. 3 is in k; 3,-3,-9,9,27,-27. 12 is not there in this series. But, it may be there.

Using both; 2,3,-2,-3,-4,-9,6,-6,12. 12 is definitely there.

Ans: "C".


Hi Fluke,

The question asks us if '12 is in K'. After writing out the sequence (as Bunnel has done), I was NOT able to get a 12. Therefore, my answer was suffcient, because I could answer definitively and say that 12 is NOT in K. I reached the same conclusion with statement (2). Therefore, my answer to this question was D.

In your quote above you mention that "[12] may be there". This is what I do not understand. If the statement given to us is a fact, then how can we assume that the sequence COULD have a 12 ? (even after we draw out the set and no 12 is present)

Bunnel, if you can, can you jump in on this one too please!




For (i) the data that is given is just enough to say that powers of 2 are present in the set.
Data is "INSUFFICIENT" to "DEFINITELY" say that 12 isn't there in the set.
It could be so that 12 is present but it hasn't been mentioned. So we can't categorically rule out the presence of 12 in the set, and say NO to the question "Is 12 in K? "

Same with (ii). The data given is "INSUFFICIENT" to say "DEFINITELY" that 12 isn't there.

Taking i and ii together, we can DEFINITELY say with this data that 12 is present. ie the statements together are SUFFICIENT
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Re: K is a set of numbers such that (i) If x is in K, then -x [#permalink] New post 15 Dec 2012, 23:54
bunuel,

This is how I solved..

For 12 to be in the set, 2 and 3 must be there.. as 12's prime factors are 2 and 3.. hence C.
Is this correct?
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Re: K is a set of numbers such that (i) If x is in K, then -x [#permalink] New post 16 Dec 2012, 07:14
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Sachin9 wrote:
bunuel,

This is how I solved..

For 12 to be in the set, 2 and 3 must be there.. as 12's prime factors are 2 and 3.. hence C.
Is this correct?


No, that's not correct. 12 can be in the set even if 2 and 3 are not.

For example, (i) says that "if x is in K, then -x is in K", then if we were told that -12 is in the set then -(-12)=12 would be in the set.
Or, (ii) say that "if each of x and y is in K, then xy is in K", then if we were told that both 2 and 6 are in the set, then 2*6=12 would be in the set.

Of course there are many other possibilities.

Hope it's clear.
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Re: K is a set of numbers such that (i) If x is in K, then -x [#permalink] New post 16 Dec 2012, 19:58
Bunuel wrote:
Sachin9 wrote:
bunuel,

This is how I solved..

For 12 to be in the set, 2 and 3 must be there.. as 12's prime factors are 2 and 3.. hence C.
Is this correct?


No, that's not correct. 12 can be in the set even if 2 and 3 are not.

For example, (i) says that "if x is in K, then -x is in K", then if we were told that -12 is in the set then -(-12)=12 would be in the set.
Or, (ii) say that "if each of x and y is in K, then xy is in K", then if we were told that both 2 and 6 are in the set, then 2*6=12 would be in the set.

Of course there are many other possibilities.

Hope it's clear.


Yeah so, for 12 to be in the set, according to the given conditions, presence of 2 of the factors (other than 1) of 12 is required .. Since 2 and 3 are present, it follows that 12 will be there..
Is this correct?
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Re: K is a set of numbers such that (i) If x is in K, then -x [#permalink] New post 16 Dec 2012, 22:22
Expert's post
Sachin9 wrote:
Bunuel wrote:
Sachin9 wrote:
bunuel,

This is how I solved..

For 12 to be in the set, 2 and 3 must be there.. as 12's prime factors are 2 and 3.. hence C.
Is this correct?


No, that's not correct. 12 can be in the set even if 2 and 3 are not.

For example, (i) says that "if x is in K, then -x is in K", then if we were told that -12 is in the set then -(-12)=12 would be in the set.
Or, (ii) say that "if each of x and y is in K, then xy is in K", then if we were told that both 2 and 6 are in the set, then 2*6=12 would be in the set.

Of course there are many other possibilities.

Hope it's clear.


Yeah so, for 12 to be in the set, according to the given conditions, presence of 2 of the factors (other than 1) of 12 is required .. Since 2 and 3 are present, it follows that 12 will be there..
Is this correct?


The red part it not correct. In the post you are quoting you can see that 12 can be there if -12 is in the set.
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Re: K is a set of numbers such that (i) If x is in K, then -x [#permalink] New post 13 Apr 2013, 07:31
Thank you Karishma. Your statement that we don't have to start with 2 or 3 to make the set was the KEY for me. It really told me that may be 12 is possible for statement 1 and 2 because earlier I thought it was ALWAYS NO and was wondering how one could get 12 in the set K. Of course, if we don't assume that we always have to start building the set with 2 or 3, then 12 is entirely possible. Since this is ALWAYS no or ALWAYS yes question, C makes much more sense now. :)

VeritasPrepKarishma wrote:
Sachin9 wrote:
bunuel,

This is how I solved..

For 12 to be in the set, 2 and 3 must be there.. as 12's prime factors are 2 and 3.. hence C.
Is this correct?


Responding to a pm:

Given in the question:

(ii) if each of x and y is in K, then xy is in K

What does this mean? It means that if x and y are in K, then xy must also be there e.g. x = 6, y = 8
If 6 and 8 are in K, 6*8 = 48 must also be in K.

Does it also mean that 2 and 3 (i.e. factors of 6) must also be in K? No. It is not necessary.
I am building the set K. I could have put 6 in on my own. I don't need to start from 2 and 3 necessarily. If 6 and 8 are in the set K, I necessarily need to put their product in too. But I needn't put in their prime factors. We do not know whether their prime factors were put in and hence 6 and 8 were obtained or whether they were put in by set maker's choice.

So, the statement 'if each of x and y is in K, then xy is in K' only implies that product of x and y must be in K too. It doesn't imply that factors of x and y must be in K. x and y could have been put in by choice. Who says that only prime factors can be added to the set? You can pick any number and add it to the set. The only thing is that once you put in that number, you must put in its product with every number already there in the set and so on...

As pointed out by Bunuel, I hope you see that your logic is not correct.
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OG-Q DS 70 [#permalink] New post 29 Oct 2013, 05:18
Can someone give me an easier explanation of the solution? thanks
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Re: OG-Q DS 70 [#permalink] New post 29 Oct 2013, 05:35
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Can someone give me an easier explanation of the solution? thanks


Merging similar topics. Please ask if anything remains unclear.

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a-set-of-numbers-has-the-property-that-for-any-number-t-in-t-98829.html
for-a-certain-set-of-numbers-if-x-is-in-the-set-then-x-136580.html
if-p-is-a-set-of-integers-and-3-is-in-p-is-every-positive-96630.html
k-is-a-set-of-integers-such-that-if-the-integer-r-is-in-k-103005.html

Hope this helps.
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Kudos [?]: 8 [0], given: 7

Re: K is a set of numbers such that (i) If x is in K, then -x [#permalink] New post 29 May 2014, 20:29
can any one tell me why (1) and (2) are insufficient ?
(1) shows that the set is …. -16,-8,-4,-,2,4,8,16….( there is no 12 here) so it is sufficient.
(2) shows that the set is …. -27,-9,-3,3,9,27,…… ( there is no 12 here) so it is sufficient.
so the answer is D … each alone is sufficient.

any explanation please? thanks
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Re: K is a set of numbers such that (i) If x is in K, then -x [#permalink] New post 29 May 2014, 21:31
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shagalo wrote:
can any one tell me why (1) and (2) are insufficient ?
(1) shows that the set is …. -16,-8,-4,-,2,4,8,16….( there is no 12 here) so it is sufficient.
(2) shows that the set is …. -27,-9,-3,3,9,27,…… ( there is no 12 here) so it is sufficient.
so the answer is D … each alone is sufficient.

any explanation please? thanks


How does statement 1 show that 12 is not in the set? All statement 1 tells you is that 2 is there and hence -2 is there. We don't know anything about other elements. How did you get 4... We are not given that if 2 is there, only powers of 2 will be there.
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Re: K is a set of numbers such that (i) If x is in K, then -x   [#permalink] 29 May 2014, 21:31
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