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# If y, z, and b are positive integers greater than 1, what is the value

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Math Expert
Joined: 02 Sep 2009
Posts: 50001
If y, z, and b are positive integers greater than 1, what is the value  [#permalink]

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02 May 2016, 22:30
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Difficulty:

75% (hard)

Question Stats:

42% (02:29) correct 58% (02:48) wrong based on 28 sessions

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If y, z, and b are positive integers greater than 1, what is the value of z?

(1) $$y^3*z^b$$ is a perfect square less than 100.
(2) $$y^3*z^b = x^3$$, where x is an integer.

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If y, z, and b are positive integers greater than 1, what is the value  [#permalink]

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03 May 2016, 08:20
y > 1; z > 1; b > 1; z = ?

St1: y^3*z^b is a perfect square less than 100
Possible Squares: 4, 9, 16, 25, 36, 49, 64, 81
Possible Cubes: 8, 27, 81

Cube * z^b = Square
Only 8 * 8 = 64 is a valid value
8 = 2^3; z^b = 2^3 --> z = 2

St2: y^3*z^b = x^3
8 * 2^3 = 4^3 --> z = 2
8 * 8^2 = 8^3 --> z = 8
Not Sufficient

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Re: If y, z, and b are positive integers greater than 1, what is the value  [#permalink]

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22 Sep 2018, 23:18
Hi chetan2u Bunuel PKN gmatbusters amanvermagmat Abhishek009 VeritasKarishma

I could not understand above solution. Could you elaborate?
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If y, z, and b are positive integers greater than 1, what is the value  [#permalink]

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23 Sep 2018, 19:19
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Hi chetan2u Bunuel PKN gmatbusters amanvermagmat Abhishek009 VeritasKarishma

I could not understand above solution. Could you elaborate?

Question stem:- We need to find out an unique value of z.

Given, y,z, and b are positive integers greater than 1.

St1:- $$y^3=$$A perfect cube*$$z^b<$$ perfect square less than 100( $$2^2, 3^2, 4^2,5^2, 6^2, 7^2, 8^2, 9^2$$)
Since y is an integer greater than 1, hence the possible values of $$y^3$$$$* z^b$$:
a) $$2^3*8=8^2=64<100$$ (A PERFECT SQUARE). KEEP
b) $$3^3*3^1=9^2$$<100( (A PERFECT SQUARE), However, b>1. DISCARD this case.

So, we have an unique value of z from (a).
Sufficient.

St2:- $$y^3*z^b=x^3$$, where x is an integer.
Here, there are no restrictions on the values of the variables x,y,z, and b.
Hence, we can have numerous values of z.

Insufficient.

Ans. (A)

P.S:- You may raise specific queries(if any)
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If y, z, and b are positive integers greater than 1, what is the value &nbs [#permalink] 23 Sep 2018, 19:19
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