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If x and y are positive integers, then what is the value of (–1)^x + (

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If x and y are positive integers, then what is the value of (–1)^x + (  [#permalink]

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New post 05 Jul 2018, 05:47
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GMAT CLUB'S FRESH QUESTION:



If x and y are positive integers, then what is the value of \((–1)^x + (–1)^y + (–1)^x*(–1)^y\) ?

(1) x does not have a prime factor greater than 1.
(2) y is a common factor of all prime numbers between 101 to 202

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If x and y are positive integers, then what is the value of (–1)^x + (  [#permalink]

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New post 05 Jul 2018, 07:17
Bunuel wrote:

GMAT CLUB'S FRESH QUESTION:



If x and y are positive integers, then what is the value of \((–1)^x + (–1)^y + (–1)^x*(–1)^y\) ?

(1) x does not have a prime factor greater than 1.
(2) y is a common factor of all prime numbers between 101 to 202


Given, x and y are positive integers

St1:- x does not have a prime factor greater than 1
So, x=1 implies\((-1)^x=(-1)^1=-1\)
when y=even, (-1)^y=1
So, \((–1)^x + (–1)^y + (–1)^x*(–1)^y\)=-1+1+(-1)(1)=-1
when y=odd, \((-1)^y=-1\)
So, \((–1)^x + (–1)^y + (–1)^x*(–1)^y\)=-1+(-1)+(-1)(-1)=-1
hence sufficient.
St2:-y is a common factor of all prime numbers between 101 to 202
So, y=1 implies \((-1)^y=(-1)^1=-1\)
when x=even, \((-1)^x=1\)
So, \((–1)^x + (–1)^y + (–1)^x*(–1)^y\)=1+(-1)+1(-1)=-1
when x=odd, (-1)^x=-1
So, \((–1)^x + (–1)^y + (–1)^x*(–1)^y\)=-1+(-1)+(-1)(-1)=-1
hence sufficient.

Ans. D
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Re: If x and y are positive integers, then what is the value of (–1)^x + (  [#permalink]

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New post 05 Jul 2018, 08:01
Bunuel wrote:

GMAT CLUB'S FRESH QUESTION:



If x and y are positive integers, then what is the value of \((–1)^x + (–1)^y + (–1)^x*(–1)^y\) ?

(1) x does not have a prime factor greater than 1.
(2) y is a common factor of all prime numbers between 101 to 202



\((–1)^x + (–1)^y + (–1)^x*(–1)^y\)

value of this expression depends on Even/Odd property of x and y , so we need to find this property of x and y

s1 the smallest Prime Number is 2, x > 0 and positive therefore x =1(Odd).
no info about y therefore not sufficient

s2 There are only two factors of any prime number, 1 and the number itself. Therefore y = 1(Odd)
no info about x therefore not sufficient.

together we have values of both x and y therefore sufficient. Answer : C
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Re: If x and y are positive integers, then what is the value of (–1)^x + (  [#permalink]

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New post 05 Jul 2018, 11:27
Bunuel wrote:

GMAT CLUB'S FRESH QUESTION:



If x and y are positive integers, then what is the value of \((–1)^x + (–1)^y + (–1)^x*(–1)^y\) ?

(1) x does not have a prime factor greater than 1.
(2) y is a common factor of all prime numbers between 101 to 202


Given x, y > 0

Asked is value of Expression E = \((–1)^x + (–1)^y + (–1)^x*(–1)^y\)

We know that (-1)^{even} = 1 & (-1)^{odd} = -1

Hence we have cases
(i) x = even, y = even, then E = 1 + 1 + 1*1 = 3

(ii) x = even, y = odd, then E = 1 - 1 + 1*(-1) = -1

(iii) x = odd, y = odd, then E = (-1) + (-1) + (-1)*(-1) = -1

(iv) x = odd, y = even, then E = (-1) + 1 + (-1)*1 = -1


Statement 1: x does not have a prime factor greater than 1. Therefore x = 1, hence x = odd

We have from cases (iii) & (iv), for x = odd, y = odd/even, E = -1

Hence statement 1 is Sufficient.


Statement 2: y is a common factor of all prime numbers between 101 to 202. Therefore y = 1, hence y = odd

We have cases (ii) & (iii), for y = odd, x = odd/even, E = -1

Hence Statement 2 is Sufficient.


Answer D.



Thanks,
GyM
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Re: If x and y are positive integers, then what is the value of (–1)^x + (  [#permalink]

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New post 06 Jul 2018, 01:14

Solution


Given:
    • x and y are positive integers in the expression \((-1)^x + (-1)^y + (-1)^x * (-1)^y\)

To find:
    • The value of the given expression

Approach and Working:
    • Considering all the three possible scenarios
      o If, x and y, both are even, then the value of the expression will be (1 + 1 + 1) = 3
      o If, x and y, both are odd, then the value of the expression will be (-1) + (-1) + 1 = -1
      o If one is even and the other is odd, then the value of the expression will be (-1) + 1 + (-1) = -1
    • Therefore, we can conclude that,
      o The value of the expression will be 3, if both x and y are even
      o The value of the expression will be -1, if at least one of them is odd

Analysing Statement 1
“x does not have a prime factor greater than 1”
    • This statement implies that x = 1 (since x does not have any prime factor)
    • Since x is odd, the value of the expression = -1 (for any value of y)

Therefore, Statement (1) ALONE is sufficient to answer this question

Analysing Statement 2
“y is a common factor of all prime numbers from 101 to 202”
    • The common factor for any two prime numbers is 1
    • Thus, the value of y is equal to 1
    • Since y is odd, the value of the expression = -1 (for any value of x)

Therefore, Statement (2) ALONE is sufficient to answer this question

Hence, the correct answer is option D.

Answer: D
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