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Math Expert V
Joined: 02 Sep 2009
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K is a set of numbers such that  [#permalink]

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Question Stats: 71% (01:27) correct 29% (01:31) wrong based on 967 sessions

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

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2
9
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.

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Hope this helps.
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Re: K is a set of numbers such that  [#permalink]

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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 ]

Originally posted by code19 on 31 Jan 2014, 00:08.
Last edited by code19 on 31 Jan 2014, 12:59, edited 4 times in total.
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Re: K is a set of numbers such that  [#permalink]

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1
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.
Math Expert V
Joined: 02 Sep 2009
Posts: 64318
Re: K is a set of numbers such that  [#permalink]

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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.

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Hope this helps.
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Re: K is a set of numbers such that  [#permalink]

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

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Done. Thank you.
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Re: K is a set of numbers such that  [#permalink]

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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:
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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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?
Math Expert V
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Re: K is a set of numbers such that  [#permalink]

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2
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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Re: K is a set of numbers such that  [#permalink]

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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
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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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.

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
Math Expert V
Joined: 02 Sep 2009
Posts: 64318
Re: K is a set of numbers such that  [#permalink]

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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.

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

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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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.

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

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

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hello,

Given that in a set of numbers K:
(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.

statement 1: 2 is in K so as per the given -2 is also there but it tell us nothing about 12 so statement 1 alone is not sufficient
statement 2: 3 is in K so as per the given -3 and 6 are also in the set K. it tells us nothing about 12.

combining statement 1 and 2 together:

since now we have 2,3,6 in the set K we can say that we also have 12 in the set K too since 2 and 6 are present in the sequence

C: both statements together are sufficient

Originally posted by GeorgeKo111 on 27 Jun 2019, 07:57.
Last edited by GeorgeKo111 on 27 Jun 2019, 08:25, edited 1 time in total.
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Re: K is a set of numbers such that  [#permalink]

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Statement 1:
a. Since 2 is in the set, -2 is definitely in the set.
b. Since 2&-2 are in the set, -4 is definitely in the set.

So, KNOWN entities in the set are 2, -2,-4,4,-16,-8,8 ........

PROBLEM IS those are the KNOWN entities and we don’t know the other (UNKNOWN) entities.
PERHAPS the set is 6,-6,2,-2,-4,12 etc (12 IS AVAILABLE)

Or
5,-5,2,-2,10,-4, -10 etc (12 IS NOT AVAILABLE)

So, STATEMENT 1 is INSUFFICIENT.

Posted from my mobile device
Current Student G
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Re: K is a set of numbers such that  [#permalink]

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Statement 2:
Here KNOWN Elements: 3,-3,-9,-27,27 etc
We don’t KNOW the UNKNOWN elements.

The set may be
4,-4,3,-3,-9,12......(12 available)
Or
5,-5, 3,-3,-9 etc (12 NOT available)

Statement 2 :insufficient .

Posted from my mobile device
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Re: K is a set of numbers such that  [#permalink]

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Combining both statements:

Known elements: 2,-2,-4,3,-3,-9,12 (YES 12 is Available in KNOWN elements).

Sufficient.

C

Posted from my mobile device Re: K is a set of numbers such that   [#permalink] 27 Jun 2019, 08:27

# K is a set of numbers such that  