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If a, b, x, y, and z are nonnegative integers and 2^a – 2^b = (3^x)(2^

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If a, b, x, y, and z are nonnegative integers and 2^a – 2^b = (3^x)(2^  [#permalink]

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New post 25 Apr 2016, 04:06
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Re: If a, b, x, y, and z are nonnegative integers and 2^a – 2^b = (3^x)(2^  [#permalink]

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New post 25 Apr 2016, 06:07
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1
Bunuel wrote:
If a, b, x, y, and z are nonnegative integers and 2^a – 2^b = (3^x)(2^y)(7^z), what is the value of xyz?

(1) a – b = 3
(2) yz > 0


Hi
Non-negative would mean O or positive integers..
\(2^a – 2^b = (3^x)(2^y)(7^z)\)..

lets see the statements--

(1) a – b = 3
\(2^a – 2^b = (3^x)(2^y)(7^z)\)..
\(2^b(2^{a-b} – 1) = (3^x)(2^y)(7^z)\)..
\(2^b(2^3 – 1) = (3^x)(2^y)(7^z)\)..
\(2^b*7 = (3^x)(2^y)(7^z)\)..
so clearly 3^x=1, since there is no 3 on LHS..
OR x=0..

now irrespective of what y and z are, xyz will remain 0..
Suff..
Otherwise we get b=y, z=1 and x=0

(2) yz > 0
Just tells us that y and z are not 0
nothing about x or the numeric values of y and z..
Insuff
A
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Re: If a, b, x, y, and z are nonnegative integers and 2^a – 2^b = (3^x)(2^  [#permalink]

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New post 27 Oct 2018, 10:35
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Ans: A
Let us start with analyzing the equation:
Since all are non-negative numbers, a>b. Which makes the question--2^b * (2^(a-b) – 1) = 3^x * 2^y * 7^z

Let us consider Statement 1:
a-b=3
Substituting gives us-> 2^b * (2^3 - 1) = 2^b * 7
So we get x=0, y=b and z=1, which gives xyz = 0.
Thus statement 1 is sufficient.

Now statement 2:
yz>0
This statement tells us that y and z both are not zero, but it tells us nothing about x.
If a-b=3, we get x=0 and thus xyz = 0.
But if a-b=6, we get x=2, y=b and z=1. Since we do not know b, we can not calculate the value of xyz.
Thus, Statement 2 is insufficient.

A is the answer.
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Re: If a, b, x, y, and z are nonnegative integers and 2^a – 2^b = (3^x)(2^  [#permalink]

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Re: If a, b, x, y, and z are nonnegative integers and 2^a – 2^b = (3^x)(2^  [#permalink]

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New post 27 Oct 2018, 10:29
a-b = 3 is insufficient
yz>0 is insufficient since there is no information about x.

2^a-2^b = 2^b(2^a-b - 1) = 2^b * 7^1 ; x=0, z= 1, y=b...
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Re: If a, b, x, y, and z are nonnegative integers and 2^a – 2^b = (3^x)(2^  [#permalink]

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Re: If a, b, x, y, and z are nonnegative integers and 2^a – 2^b = (3^x)(2^  [#permalink]

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New post 25 Dec 2018, 01:05
Bunuel wrote:
If a, b, x, y, and z are nonnegative integers and \(2^a – 2^b = (3^x)(2^y)(7^z)\), what is the value of xyz?


(1) \(a – b = 3\)

(2) \(yz > 0\)



from 1:

2^b(2^a-b -1 )= 3^x*2^y*7^z

given a-b=3
2^b(7)=3^x*2^y*7^z

3^x=1 or say x=0
so xyz= 0

sufficeint

from 2
yz>0
no relation given in terms of a,b,x so in sufficeint

IMO A
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Re: If a, b, x, y, and z are nonnegative integers and 2^a – 2^b = (3^x)(2^   [#permalink] 25 Dec 2018, 01:05
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