If \(x\) and \(y\) are integers and \(x \lt y\), what is the value of \(x + y\)?


(1) \(x^y = 4\)

(2) \(|x| = |y|\)
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Official Solution:


The question cannot be easily rephrased to incorporate the particular information given. However, of course we should take note that both variables are integers and that \(x\) is less than \(y\). We are looking for the value of \(x + y\).

Statement (1): SUFFICIENT. First, we should list out all the possible scenarios in which integers \(x\) and \(y\) fit the equation \(x^y = 4\).

There are three possibilities, as we can find by trial and error: \(2^2 = 4\), \((-2)^2 = 4\), and \(4^1 = 4\). However, of these possibilities, there is only one for which \(x\) is less than \(y\), namely \((-2)^2 = 4\). Thus, we can find the value of \(x + y\), which is \(-2 + 2 = 0\).

Statement (2): SUFFICIENT. Knowing that \(|x| = |y|\) does not tell us the values of the integers. However, since they have the same absolute value, but \(x\) is less than \(y\), it must be the case that \(y\) is a positive integer and \(x\) is the negative of that integer. For instance, if \(y\) is 5, then \(x\) is -5. The sum of \(x\) and \(y\) must therefore be 0, no matter what.


Answer: D
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I didn't understand the logic of the solution. By knowing |x| = |y|, how can we find out x+y? Also, when it is mentioned that x<y.

Please read the first sentence of the question: If \(x\) and \(y\) are integers and \(x \lt y\),...
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Thanks.. I got the point.

I still don't understand how the second statement in sufficient.

If x and y are integers and x<y, what is the value of x+y?

(1) x^y=4 --> as x and y are integers and x<y then only possible solution is (-2)^2=4 (other integer solutions for x^y=4 are: 2^2=4 and 4^1=4) --> x+y=-2+2=0. Sufficient.

(2) |x|=|y| --> as also x<y then they have opposite signs (x<0<y, so |x|=-x and|y|=y) --> -x=y --> x+y=0. Sufficient.

Answer: D.

Hope it helps.
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I love this question, got it wrong because I didnt understand the logic of statement 2. Cheers

I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation. interesting ques, though got it wrong(chose option A) but learned a point.
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i had the question wrong but I really appreciate the solution

I think this is a high-quality question and I agree with explanation.

If x and y are integers and x<y, what is the value of x+y?

(1) \(x^y\)=4
The only number that when squared gives 4 is 2. \(2^2\) = 4, but since x < y, the only option we have is \(-2^2\)

x=-2 and y=2, so x+y = 0
Sufficient

(2) |x|=|y|
With this option we know that x and y are same numbers and from x < y, we know that they are opposite in sign. So x+y =0
Sufficient

Ans D
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I think this is a poor-quality question and I don't agree with the explanation. It is given in the question that x<y, We can't assume either x=2 and y=2, or x=4 and y=1. I believe this breaches the question statement. Let me know if I have wrong understanding.

The solutions above do take into account that x < y. No?
https://gmatclub.com/forum/s96-184673.html#p1750025
https://gmatclub.com/forum/s96-184673.html#p1416014
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I think this is a high-quality question and I agree with explanation.

A very nice question