What is the value of xy?

Question

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

1)    -1 < –x^y < 0

2)    -10 < y^x < -1

shameekv1989 · Accepted Answer

Answer should be C IMO Since we are given x and y are integers. From i) x has to be positive and y has to be negative so that 1 < -x^y < 0 => x can not be 1 x=3; y=-2 => -x^y = -3^{-2} = \frac{-1}{9} which is between 0 and -1 x=3; y=-3 => -x^y = -3^{-3} = \frac{-1}{27} which is between 0 and -1 2 Values - Insufficient From ii) x= 3; y=-2 y^x = -2^3 = -8 which is between -1 and -10 x=1; y=-3 y^x = -3^{1} = -3 which is between -1 and -10 2 Values - Insufficient Combining i) and ii) We can not have x = 1 that restricts our values for x in statement ii) Moreover x can not be even i.e. 2 since it will make a value in statement ii) positive. In addition to all of this the max value that x can take will be 3 when y is -2 so that both statements are satisfied. xy will have a unique value then = -6 Answer - C

shanks2020 · Answer

It definitely takes more than 2 minutes. any tricks to do these kind of problems faster?

BrentGMATPrepNow · Answer

Given : x and y are integers

Target question :    What is the value of  xy ?

Statement 1 :  -1 < –x^y < 0  
This is the same as writing:  -1 < –(x^y) < 0

For now, let's just consider the fact that  -(x^y)  is negative. 
This means  x^y  must be POSITIVE
This means EITHER x is positive, and y has any value OR x is negative and y is EVEN

Now let's also consider the fact that  -1 < –(x^y) < 0 
This means  0 < (x^y) < 1 
We can now conclude that x is POSITIVE, and y is some NEGATIVE integer.

Given this, here are some possible cases that satisfy statement 1: 
 Case a : x = 2 and y = -2. In this case, the answer to the target question is  xy = (2)(-2) = -4 
 Case b : x = 2 and y = -3. In this case, the answer to the target question is  xy = (2)(-3) = -6 
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 :  -10 < y^x < -1  
If  y^x  is negative, then we know that y must be NEGATIVE and x must be ODD
Given this, here are some possible cases that satisfy statement 2: 
 Case a : x = 1 and y = -2. In this case, the answer to the target question is  xy = (1)(-2) = -2 
 Case b : x = 1 and y = -3. In this case, the answer to the target question is  xy = (1)(-3) = -3 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
Statement 1 tells us that x is POSITIVE, and y is some NEGATIVE integer. 
Statement 2 tells us that x must be ODD and y must be NEGATIVE

In order for  -10 < y^x < -1  and for  -1 < –x^y < 0 , it MUST be the case that y = -2 and x = 3
The answer to the target question is  xy = (3)(-2) = -6 
Since we can answer the  target question  with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

CrackverbalGMAT · Answer

The fact that x and y are given to be integers is a great thing because it reduces the number of cases we will have to try. To find the value of xy, we will need unique values for x and y.

From statement I alone, -1<- x^y <0. The statement tells us that the value of - x^y  is a negative proper fraction. This is possible only if x is a positive value and y is a negative odd integer. Additionally, x cannot be 1 since  -1^{odd}  = -1 which is not permissible.

Examples of possible values are as follows:
If x = 2 and y = -3, - x^y  = - 2^{-3}  = - \frac{1}{8}  which is in the given range. The value of xy is -6.
If x = 3 and y = -5, - x^y  = - 3^{-5}  = - \frac{1}{243}  which is in the given range. The value of xy is -15. 
Statement I alone is insufficient. Answer options A and D can be eliminated, possible answer options at this stage are B, C or E.

From statement II alone, we know that -10< y^x <-1. This can happen only if y is negative and x is positive. y cannot be equal to 1 whereas x can be equal to 1.
If x = 3 and y = -2,  y^x  = -8 which is in the given range. The value of xy is -6.
If x = 1 and y = -2,  y^x  = -2 which is in the given range. The value of xy is -2.
Statement II alone is insufficient. Answer option B can be eliminated, possible answer options at this stage are C or E.

Combining statements I and II, we have the following:
From statement I, we know that x cannot be equal to 1. When we apply this condition on the expression and the range given in statement II, we see that the only set of values that satisfy both statements together is x = 3 and y = - 2. The value of xy = -6.
The combination of statements is sufficient. Answer option E can be eliminated.

The correct answer option is C .

In such questions, there is no defined start point to the solution. The best thing to do in such cases is to start of by plugging in simple values which will satisfy the given constraints. Overspending time on determining what could be an optimal approach is a bane in problems like these.

Hope that helps!

meetsagar · Answer

I marked  E  as there can be multiple combinations of integer/ non integer solutions of (x,y)

Why are we assuming x & y to be integers for the given problem?

BrentGMATPrepNow · Answer

The question states that x and y are integers.

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What is the value of xy?

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