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

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.

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Hope this helps.
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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?

(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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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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Sorry but your explanation is not clear.
and i got these numbers " -16,-8,-4,-,2,4,8,16…" after applying the statement (1) on the equation (i) and (ii).
what i understood is that if you applied statement (1) "2 in K", then you will get this set of numbers " -16,-8,-4,-,2,4,8,16…" which do not include 12. the same apply on statement (2).
could you explain what is the flaw in my understanding.

Thanks

Given:
(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
Stmnt 1: 2 is in K

This implies 2, -2, -4 (2*-2), -8 (2*-4), 8 (-2*-4), etc are in the set. 4 will not be there. But how do you know that no other elements are there in the set? Could we have a set like this: {12, 2, 24, -2, -12, -24 ...}
Does it satisfy all 3 conditions given above? Yes.
The set {2, -2, -4, -8, ...} satisfies all 3 conditions too.

Hence we don't know what the set actually looks like. Just because the set has 2 doesn't mean it cannot have 12.
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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?

(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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Solution:

Statement One Only:

2 is in K.

Since 2 is in K, -2 is in K. Furthermore, 2 * -2 = -4 and 2 * -4 = -8 and -2 * -4 = 8 (and so on) are in K. We see that these numbers are powers of 2 and their opposites. If these are all the numbers in K, then 12 is not in K. However, if 3 is in K, then 4 * 3 = 12 is in K. Statement one alone is not sufficient.

Statement Two Only:

3 is in K.

Since 3 is in K, -3 is in K. Furthermore, 3 * -3 = -9 and 3 * -9 = -27 and -3 * -9 = 27 (and so on) are in K. We see that these numbers are powers of 3 and their opposites. If these are all the numbers in K, then 12 is not in K. On the other hand, if -4 is also in K, then -3 * -4 = 12 is in K. However, since we don’t know that, statement two alone is not sufficient.

Statements One and Two Together:

We see that -4 is in K (from statement one) and -3 is in K (from statement two), so -4 * -3 = 12 is in K.


Answer: C

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Why can't we say that NO 12 is not in the set K considering each option alone?

From (1) we know that 2, -2, -4, 4, 8, -8, 16, -16, ... (powers of 2 and their negative pairs) are in K but this does not mean that there are no other numbers in the K. There CAN be ANY other numbers in K, we don't know. You can notice that other numbers could also be in K if you look at (2), which says that 3 is in K. We don't get that 3 is in K in (1) while (2) says that it is. If 3 could be in K why not 12?
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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.
Insufficient.
2,-2 are in K

(2) 3 is in K.
Insufficient.
3,-3 are in K

C:
2 and 3 are in K so 2 x 3 = 6 is in K
-2 and -3 are in K so -2 x -3 = 6 is in K
2 and 6 are in K so 2 x 6 = 12 is in K

Sufficient
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KarishmaB that is because 2*-2=-4 and for every x, there will be -x and hence if we have -4 then we will also have -(-4). Isn't it. Do you still think 4 cannot be there? Pls clarify ThatDudeKnows avigutman

The set has 4 in it but we cannot say that the set is exactly {…. -16,-8,-4,-,2,4,8,16…}. These are some of the values but the set may be bigger than this.
We are implying here that there are some values less than -16, then -8 (no values between -8 and -16), then -4, then -2, then 2, then 4, then 8, then 16 (not values between them). Then we have some values greater than 16. We cannot say this.
If 2 is there, we can say that -2/-4/4 etc are there but we cannot say that 12 is NOT there.
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KarishmaB

Hope that you had a great weekend!

I immediately thought of this problem below when I saw this question:
https://gmatclub.com/forum/a-set-of-num ... 60975.html

I originally chose answer E for this problem.

For the problem that I linked to, you cannot use the xy rule to work backward (so you cannot deduce that 2*3=6... in other words, you cannot work backward based on the xy rule). Whereas, for this problem, you can use the xy rule to work forward?

Many thanks

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