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K is a set of numbers such that

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K is a set of numbers such that [#permalink]

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

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.

Data Sufficiency
Question: 70
Category: Arithmetic Properties of numbers
Page: 158
Difficulty: 600


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SOLUTION

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 --> 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.

Similar questions to practice:
for-a-certain-set-of-numbers-if-x-is-in-the-set-then-both-161920.html
for-a-certain-set-of-numbers-if-x-is-in-the-set-then-x-136580.html
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if-p-is-a-set-of-integers-and-3-is-in-p-is-every-positive-96630.html
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k-is-a-set-of-integers-such-that-if-the-integer-r-is-in-k-103005.html

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Re: K is a set of numbers such that [#permalink]

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New post 31 Jan 2014, 01:08
Let's analyze statements with what's given in the question.

Statement (1)
2 is in K
(i) -2 is also in K
(ii) 2 and -2 are both in K, so -4 is also in K .... then +4... and then -8 and +8 are in K ... looks like 2^n and -2^n are included where n is integer
which does not include 12 definitely...
So Statement (1) is sufficient to answer the question "Is 12 in K?", answer being no.


Statement (2)
Similarly results in a set 3^n and -3^n , which again answers our question, that 12 is definitely not part of the set K.
So Statement (2 is sufficient to answer the question "Is 12 in K?", answer being no.

Answer D; Both statement 1 & statement 2 are ALONE sufficient.





[oops It turns out my answer was wrong ... Just leaving the post as it is, so you know what not to do :)]
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Last edited by code19 on 31 Jan 2014, 13:59, edited 4 times in total.

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Re: K is a set of numbers such that [#permalink]

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Ans. C
From S1:if 2 is in the series,then -2 will also be there.
And if 2 & -2 are there -4 will be there.If -4 is in the series, 4 will also be there...and so on
The series becomes:2,-2,4,-4...powers of 2
But the stimulus remains silent about what is not there in this series.So insufficient.(12 might or might not be there.)

Same explanation for S2:The series will have numbers with powers of 3.

Together for S1 & S2,at some point we'll have multiple of 3 and 4 because if 3 and 4 are there in the series,their multiple will definitely be there as implied by the second statement in stimulus.Sufficient.

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SOLUTION

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 --> 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.

Similar questions to practice:
for-a-certain-set-of-numbers-if-x-is-in-the-set-then-both-161920.html
for-a-certain-set-of-numbers-if-x-is-in-the-set-then-x-136580.html
a-set-of-numbers-has-the-property-that-for-any-number-t-in-t-98829.html
if-p-is-a-set-of-integers-and-3-is-in-p-is-every-positive-96630.html
k-is-a-set-of-numbers-such-that-i-if-x-is-in-k-then-x-96907.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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Re: K is a set of numbers such that [#permalink]

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New post 19 Feb 2014, 07:01
Bunuel, one quick query -> When we say that (from stmt 1) 2 is there in the set and hence -2 is also there -> Here we take 2 and -2 as x and -x, but then we also apply the logic x*y = -4 (here we consider 2 as x and -2 as y (and not as -x)).
Could there be a flaw in the problem statement?


Bunuel wrote:
pradeepss wrote:
Bunel can you update the oa? It shows as d on gmat timer.

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New post 19 Feb 2014, 07:52
sunnymon wrote:
Bunuel, one quick query -> When we say that (from stmt 1) 2 is there in the set and hence -2 is also there -> Here we take 2 and -2 as x and -x, but then we also apply the logic x*y = -4 (here we consider 2 as x and -2 as y (and not as -x)).
Could there be a flaw in the problem statement?


Bunuel wrote:
pradeepss wrote:
Bunel can you update the oa? It shows as d on gmat timer.

________________
Done. Thank you.


(i) and (ii) are general rules for the set, meaning that they apply to any numbers in the set:

(i) if a number is in K, then - that number is also in K
(ii) for any two numbers in the set, their product is also in the set.

Hope it's clear.
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K is a set of numbers such that [#permalink]

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New post 22 Mar 2015, 03:23
What is unclear to me is that it states that xy is in the set, but how can we infer that x*x and y*y is in the set.

-2 and 2
-3 and 3 gives us 6? How can do we infer that -2 * 2 = 4 is in the set?

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erikvm wrote:
What is unclear to me is that it states that xy is in the set, but how can we infer that x*x and y*y is in the set.

-2 and 2
-3 and 3 gives us 6? How can do we infer that -2 * 2 = 4 is in the set?


(i) and (ii) are general rules for the set, meaning that they apply to any numbers in the set:

(i) if a number is in K, then - that number is also in K
(ii) for any two numbers in the set, their product is also in the set.

(1) says that 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.

(2) says 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.

Does this make sense?
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K is a set of numbers such that [#permalink]

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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


Can you please explain the solution?

Last edited by Vyshak on 29 Oct 2016, 09:25, edited 1 time in total.
Topic Merged. Refer to the above discussions

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K is a set of numbers such that [#permalink]

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Oops - sorry. Looks like I was responding while Bunuel was merging topics.
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Re: K is a set of numbers such that [#permalink]

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New post 23 Feb 2017, 04:52
PROMPT ANALYSIS

Let us say that x and y are part of K
Hence K ={x,y,-x,-y,-xy,-y*y, -x*x….}

SUPERSET
The answer will be either YES or NO.

TRANSLATION
To know the answer
1# exact value of the element of K
2# some values to calculate the rest of the value

STATEMENT ANALYSIS

St 1: if 2 is in K, the -2 is in K, -4 is in K and so on. Hence, all the calculated elements will be in the form of 2n.we cannot say about rest of the element. Option a and d eliminated

St 2: if 3 is in K, the -3 is in K, -9 is in K and so on. Hence, all the calculated elements will be in the form of 3n.we cannot say about rest of the element.option b eliminated

St 1 & St 2: If 2 and 3 is in K, -2 and -3 is in K, -4 is also in K, 12 is in K.ANSWER

Option C

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Re: K is a set of numbers such that [#permalink]

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New post 06 May 2017, 09:18
Bunuel wrote:
SOLUTION

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 --> 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 this helps.


Hi Bunuel (or anyone else who can explain this),

In the sets mentioned above:
2, -2, -4, 4, 8, -8, 16, -16, ...
3, -3, -9, 9, 27, -27, 81, -81, ...

How are we getting 8, -8 in the first set and 27, -27 in the 2nd? Shouldn't it be 2,-2,-4,4,-16,16... and 3,-3,-9,9,-81,81...

The reason is that if we are taking xy as x and -x in statement (i), how do we get (for example) 8,-8 if we get -4 and 4 for x and -x? Is it that we are keeping y constant at 2 throughout? Please clarify.

Thanks.

Graeme

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Re: K is a set of numbers such that [#permalink]

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New post 06 May 2017, 10:11
Graeme520 wrote:
Bunuel wrote:
SOLUTION

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 --> 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 this helps.


Hi Bunuel (or anyone else who can explain this),

In the sets mentioned above:
2, -2, -4, 4, 8, -8, 16, -16, ...
3, -3, -9, 9, 27, -27, 81, -81, ...

How are we getting 8, -8 in the first set and 27, -27 in the 2nd? Shouldn't it be 2,-2,-4,4,-16,16... and 3,-3,-9,9,-81,81...

The reason is that if we are taking xy as x and -x in statement (i), how do we get (for example) 8,-8 if we get -4 and 4 for x and -x? Is it that we are keeping y constant at 2 throughout? Please clarify.

Thanks.

Graeme


First of all please read this: https://gmatclub.com/forum/k-is-a-set-o ... l#p1503835

Next:
(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 --> according to (ii) 2*4 = 8 is in K --> according to (i) -8 is in K...

Hope it's clear.
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Re: K is a set of numbers such that [#permalink]

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New post 06 May 2017, 10:36
Bunuel wrote:
Graeme520 wrote:
Bunuel wrote:
SOLUTION

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 --> 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 this helps.


Hi Bunuel (or anyone else who can explain this),

In the sets mentioned above:
2, -2, -4, 4, 8, -8, 16, -16, ...
3, -3, -9, 9, 27, -27, 81, -81, ...

How are we getting 8, -8 in the first set and 27, -27 in the 2nd? Shouldn't it be 2,-2,-4,4,-16,16... and 3,-3,-9,9,-81,81...

The reason is that if we are taking xy as x and -x in statement (i), how do we get (for example) 8,-8 if we get -4 and 4 for x and -x? Is it that we are keeping y constant at 2 throughout? Please clarify.

Thanks.

Graeme


First of all please read this:

Next:
(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 --> according to (ii) 2*4 = 8 is in K --> according to (i) -8 is in K...

Hope it's clear.



Thanks for responding. Think I got it.

Therefore, we keep the pattern, substituting xy found in (ii) into "x" found in (i) but then, when coming back to (ii), leaving the original "x" = 2 in xy. Correct?

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Re: K is a set of numbers such that   [#permalink] 06 May 2017, 10:36
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