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# If @ represents one of the operations +, -, and x, is k@(l+m

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Re: If @ represents one of the operations +, -, and x, is k@(l+m [#permalink]
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Bunuel
If @ represents one of the operations +, -, and x, is k@(l+m)=(k@l)+(k@m) for all numbers k, l,and m?

(1) k@1 is not equal to 1@k for some numbers k. @ is neither addition (as $$k+1=1+k$$) nor multiplication (as $$k*1=1*k$$), thus @ represents subtraction. Knowing that we can determine whether $$k-(l+m)=(k-l)+(k-m)$$ for all numbers k, l,and m. Sufficient.

(2) @ represents subtraction. The same here. Sufficient.

Dear Bunnel,
I would like to understand the above question first..
If we take the @ as subtraction from statement 1 and 2 then the equation stands as $$k-l-m=2k-l-m$$, which is not equal in both the side.

I was wondering whether the question asks about the operation of the @ sign, which makes the equation of k@(l+m)=(k@l)+(k@m) okay from both end.

Thanks
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Re: If @ represents one of the operations +, -, and x, is k@(l+m [#permalink]
Drik
Bunuel
If @ represents one of the operations +, -, and x, is k@(l+m)=(k@l)+(k@m) for all numbers k, l,and m?

(1) k@1 is not equal to 1@k for some numbers k. @ is neither addition (as $$k+1=1+k$$) nor multiplication (as $$k*1=1*k$$), thus @ represents subtraction. Knowing that we can determine whether $$k-(l+m)=(k-l)+(k-m)$$ for all numbers k, l,and m. Sufficient.

(2) @ represents subtraction. The same here. Sufficient.

Dear Bunnel,
I would like to understand the above question first..
If we take the @ as subtraction from statement 1 and 2 then the equation stands as $$k-l-m=2k-l-m$$, which is not equal in both the side.

I was wondering whether the question asks about the operation of the @ sign, which makes the equation of k@(l+m)=(k@l)+(k@m) okay from both end.

Thanks

No, the question asks: "is k@(l+m)=(k@l)+(k@m) for ALL numbers k, l,and m", where @ represents one of the operations +, -, and x.
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Re: If @ represents one of the operations +, -, and x, is k@(l+m [#permalink]

If we take k=l=m=0 & k=1, l=2, m=3, from statement 1 we will get both "yes" or "no". Similarly Statement 2 also gives the same result.

The question doesn't specify anything about k,l,m

Could you explain?
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Re: If @ represents one of the operations +, -, and x, is k@(l+m [#permalink]
The way I approached this question was basically "is @ multiplication"?

That is the only symbol that will make the equation in the question stem equal.

The first statement tells us indeed that @ is not multiplication or even addition. The only other option is subtraction...so we have our answer and it is not multiplication. Sufficient.

The second statement tells us @ is subtraction. Ok so we know it is not multiplication. Sufficient.
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Re: If @ represents one of the operations +, -, and x, is k@(l+m [#permalink]
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The way i see the question,
k o (l +m) = (k o l) + (k o m) is only true where o is x(multiplication)
for o = + and o = -, it's not true.

1. k o 1 not equal to 1 o k. This statement is true only when o is subtraction (-). But we know that the above statement is valid only for multiplication. So this option is SUFFICIENT.
2. o represents subtraction . This statement is SUFFICIENT , as we know that the question is valid only for multiplication.
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Re: If @ represents one of the operations +, -, and x, is k@(l+m [#permalink]
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If @ represents one of the operations +, -, and x, is k@(l+m)=(k@l)+(k@m) for all numbers k, l,and m?

(1) k@1 is not equal to 1@k for some numbers k.
(2) @ represents subtraction.

The answer to this question could be a yes or a no. If we can somehow say for sure - yes or no, then we know the option is sufficient.
2) clearly says @ is subtractn. Therefore, the equation in the question is NOT true for all nos. k,l,m. SUFFICIENT.

1)k@1 != 1@k implies that @ s not x . This could be + since if k is neg, -k+1 is not equal to 1-(-k)
This could be - since k-1 != 1-k.
substituting in the question, for +: is k+(l+m)=(k+l) + (k+m). NO.
for - : is k-(l-m)= (k-l) + (k-m) . NO.

There the equation is NOT true for all nos. SUFFICIENT.

D it is!
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Re: If represents one of the operations +, -, and x, is k(l+m [#permalink]
St1 : @ is substraction

if we take K=5, l=3, m=2
Then 0 is not equal to 5....as per question stem answer is NO.

If we take K=0,l=1,m= 2

Then -3= -3....so answer is YES

Since we get yes and no both....shouldn't this statement be insufficient ?

Where am I doing the mistake ?
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If represents one of the operations +, -, and x, is k(l+m [#permalink]
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This is a tricky question that troubles a lot of students.

Based on statement 1, the symbol {o} has to be equal to subtraction, because in the case of addition and multiplication the operation k{o}1 is equal to 1{o}k for all all numbers. So the conclusion from statement 1 is that the symbol stands for only subtraction.

Now if we go back to the original question in the main stem, which asks if Does k{o}(l+m) = (k{o}l) + (k{o}m) for all numbers k, l, and m? meaning is the answer to this question a definite Yes or No. If the {o} stands for subtraction then the condition k-{l+m} is not equal to (k-l) + (k-m) for all numbers. It may hold true for k=l=m=0, but we need to answer the question if it holds true for all possible values of k, l, and m, and the answer to that is a definite No, which makes it sufficient.

Cheers,
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Re: If @ represents one of the operations +, -, and x, is k@(l+m [#permalink]
Top Contributor
Solution:

St(1):-k@1 is not equal to 1@k for some numbers k.

Commutative property is not satisfied

The operator represents "-"

The rephrased question stem is is k-(l+m)=(k-l)+(k-m) ?

Only at k=0 the question holds good and thus answer is a NO for all numbers

St(2):-Same logic (Sufficient)

(option d)

Devmitra Sen
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Re: If @ represents one of the operations +, -, and x, is k@(l+m [#permalink]
CrackverbalGMAT
Solution:

St(1):-k@1 is not equal to 1@k for some numbers k.

Commutative property is not satisfied

The operator represents "-"

The rephrased question stem is is k-(l+m)=(k-l)+(k-m) ?

Only at k=0 the question holds good and thus answer is a NO for all numbers

St(2):-Same logic (Sufficient)

(option d)

Devmitra Sen
GMAT SME

Thanks CrackverbalGMAT for great explanation. What is the purpose of testing k=0 here and how would we know the requirement to test k=0?
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Re: If @ represents one of the operations +, -, and x, is k@(l+m [#permalink]
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