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If x, y, and z are integers, is x even?

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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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Posts: 1
 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?

 Post subject: Re: If x, y, and z are integers, is x even?  Posted: Mon Jun 04, 2012 1:03 am
 Course Students

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 Exponent simplification.Is X even?10^x= 4^y * 5^zfirst off "10^x"="(2^X)(5^X)"Statement 1 is a FACT. Put it in this form to see if x is even4^y= 2^2^y= 2^2y 5^z is simplified so the rephrase is10^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. SUFF2)3^x+5=27^y+1always simplify3^x+5= 3^3(y+1)x+5= 3y+3subtract 5x=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. INSUFFA

 Post subject: Re: If x, y, and z are integers, is x even?  Posted: Sat Jun 09, 2012 10:57 pm
 ManhattanGMAT Staff

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 Thanks! pshu4, please let us know if you need further assistance. :-) _________________Jamie NelsonManhattanGMAT Instructor

 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?

 Post subject: Re: If x, y, and z are integers, is x even?  Posted: Tue Feb 05, 2013 4:35 pm
 ManhattanGMAT Staff

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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 SandersManhattan GMAT Instructor

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