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pshu4
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Post subject: If x, y, and z are integers, is x even?  Posted: Sun Jun 03, 2012 7:16 pm |
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This is from the Number Properties book 5th edition, pg 81, question 1.
If x, y, and z are integers, is x even?
(1) 10^x = (4^y)(5^z)
(2) 3^(x+5) = 27^(y+1)
I am confused as to how the book solves (1). How do you do it?
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ShyT
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Post subject: Re: If x, y, and z are integers, is x even? Posted: Mon Jun 04, 2012 1:03 am |
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Exponent simplification.
Is X even?
10^x= 4^y * 5^z
first off "10^x"="(2^X)(5^X)"
Statement 1 is a FACT. Put it in this form to see if x is even
4^y= 2^2^y= 2^2y
5^z is simplified so the rephrase is
10^x= (2^2y)(5^Z)
for this statement to be true. remember it is true. its a stem.. x=2y=z, 2y is even so x is even z is even. SUFF
2)3^x+5=27^y+1
always simplify
3^x+5= 3^3(y+1)
x+5= 3y+3
subtract 5
x=3y-2
When you become seasoned with your number properties you will immediately notice that this rephrased statement is insufficient because depending on "y" x can be either odd or even. If you can't see it though simply plug in numbers and you'll see x can be either odd or even. INSUFF
A
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jnelson0612
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Post subject: Re: If x, y, and z are integers, is x even? Posted: Sat Jun 09, 2012 10:57 pm |
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Thanks! pshu4, please let us know if you need further assistance. :-)
_________________
Jamie Nelson
ManhattanGMAT Instructor
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asingleton08
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Post subject: Re: If x, y, and z are integers, is x even? Posted: Mon Feb 04, 2013 10:37 pm |
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I'm sorry but I still don't understand the answer to this question:
Here's what I do "understand" so please correct if I'm wrong.
Statement 1 says 10^x=(4^y)(5^z)
So "10^x"="(2^X)(5^X)"- got that but why does that break down make x an even?
I get that 4^y=2^2y but how does that apply to:
" 5^z is simplified so the rephrase is:
10^x= (2^2y)(5^Z)
for this statement to be true. remember it is true. its a stem.. x=2y=z, 2y is even so x is even z is even. SUFF"
Especially how did you find out x=2y and x=z?
Can someone provide a more detailed explanation?
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tim
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Post subject: Re: If x, y, and z are integers, is x even? Posted: Tue Feb 05, 2013 4:35 pm |
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Posts: 4404
Location: Southwest Airlines, seat 21C
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it sounds like you understand that we have (2^2y)(5^z) = (2^x)(5^x). from this (because we're dealing with integers) we can conclude that 2^2y = 2^x, so 2y = x, and since x is two times an integer y, that makes x even. does this help?
_________________
Tim Sanders
Manhattan GMAT Instructor
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