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If x and y are integers and x^y = 256, what is the number of the diffe

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If x and y are integers and x^y = 256, what is the number of the diffe  [#permalink]

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New post 15 Nov 2019, 01:25
00:00
A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

29% (01:52) correct 71% (02:05) wrong based on 56 sessions

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Location: India
Concentration: Entrepreneurship, Marketing
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Re: If x and y are integers and x^y = 256, what is the number of the diffe  [#permalink]

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New post 15 Nov 2019, 01:43
1
Bunuel wrote:
If x and y are integers and \(x^y = 256\), what is the number of the different values of \(y^x\)?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Are You Up For the Challenge: 700 Level Questions


\(x^y = 256 = 2^8, (-2)^8, 4^4, (-4)^4, 16^2, (-16)^2, 256^1\)
--> Possible values of (x, y) = {(2, 8), (-2,8), (4, 4), (-4, 4), (16, 2), (-16, 2), (256, 1)}
--> Number of different values of \(y^x = {8^2, 8^{-2}, 4^4, 4^{-4}, 2^{16}, 2^{-16}, 1^{256}}\) = 7 values

IMO Option D
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Re: If x and y are integers and x^y = 256, what is the number of the diffe  [#permalink]

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New post 15 Nov 2019, 03:56
2
2
Bunuel wrote:
If x and y are integers and \(x^y = 256\), what is the number of the different values of \(y^x\)?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Are You Up For the Challenge: 700 Level Questions

\(x^y = 256\)
x=2 and y = 8
so \(y^x\) = 8^2 ; 64
64 = 2^6 ; total factors = 7
IMO D
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Re: If x and y are integers and x^y = 256, what is the number of the diffe  [#permalink]

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New post 15 Nov 2019, 04:22
1

Solution



Given
    • x^y=256

To find
    • Number of the different values of y^x

Approach and Working out

    • We know that 2^8 = 256
      o Hence, x= 2 and y =8
         y ^ x = 8^2 = 64

    • We know that -2^8 = 256
      o Hence, x= -2 and y =8
         y ^ x = 8^-2 = 1/64

    • 4^4 = 256
      o Hence, x= 4 and y =4
         y ^ x = 4^4 = 256

    • -4^4 = 256
      o Hence, x= -4 and y =4
         y ^ x = 4^-4 = 1/256

    • 16^2 = 256
      o Hence, x= 16 and y =2
         y ^ x = 2^16


    • -16^2 = 256
      o Hence, x= -16 and y =2
         y ^ x = 1/2^16

    • 256^1 = 256
      o Hence, x= 256 and y =1
         y ^ x = 1^256 = 1

So, total 7 values are possible.

Thus, option D is the correct answer.
Correct Answer: Option D
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Re: If x and y are integers and x^y = 256, what is the number of the diffe  [#permalink]

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New post 15 Nov 2019, 05:15
I doubt that explanation is correct though your answer choice is fortunately correct.

For example.
2 is one of the factor of 64, but 2 is not the value of y^x in any case.

Archit3110 wrote:
Bunuel wrote:
If x and y are integers and \(x^y = 256\), what is the number of the different values of \(y^x\)?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Are You Up For the Challenge: 700 Level Questions

\(x^y = 256\)
x=2 and y = 8
so \(y^x\) = 8^2 ; 64
64 = 2^6 ; total factors = 7
IMO D
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Joined: 20 Jul 2017
Posts: 1139
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
Re: If x and y are integers and x^y = 256, what is the number of the diffe  [#permalink]

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New post 22 Nov 2019, 00:42
Archit3110 wrote:
Bunuel wrote:
If x and y are integers and \(x^y = 256\), what is the number of the different values of \(y^x\)?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Are You Up For the Challenge: 700 Level Questions

\(x^y = 256\)
x=2 and y = 8
so \(y^x\) = 8^2 ; 64
64 = 2^6 ; total factors = 7
IMO D


Archit3110

How are factors and values of \(y^x\) same ?
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Re: If x and y are integers and x^y = 256, what is the number of the diffe   [#permalink] 22 Nov 2019, 00:42
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