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

Question

(1)  10^x = 4^y*5^z

(2)  3^{(x + 5)} = 27^{(y + 1)}

GMATinsight · Accepted Answer

Question : Is x Even?

Statement 1:  10^x = 4^y*5^z

i.e.  2^x * 5^x = 2^{2y} * 5^z

i.e.  x = 2y = z 
Since, y is an Integer and  x=2y , therefore x must be a multiple of 2 i.e. an Even Integer

Hence   SUFFICIENT

Statement 2:  3^{(x + 5)} = 27^{(y + 1)}

i.e.  3^{(x + 5)} = 3^{(3y + 3)}

i.e.  (x+5) = (3y+3) 
i.e.  x = 3y - 2 
But x will be odd if y is odd
and x will be even if y is even
 Hence   NOT SUFFICIENT

Answer: Option  A

ManojReddy · Answer

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

(1) 10^x = 4^y*5^z

2^x * 5^x = 2^2y * 5^z
equating, we get
x=2y ( x will be even regardless whether y is even or odd )
x=z (  x will be even if z is even and x will be odd if z is odd )

Insufficient

(2) 3^(x + 5) = 27^(y + 1)

3^(x + 5) = 3^3(y + 1)
equating, we get

x + 5 = 3(y + 1)
x = 3y - 2 ( x will be even if y is even and x will be odd if y is odd )

Insufficient

Combining statements 1 and 2

we have below equations
x=2y ------(1)
x=z --------(2)
x=3y-2 ----(3)

from (1) and (3)
2y = 3y-2
y=2 (y is even)

so x will be even

Sufficient.

z will also be even

Ans:C

GMATinsight · Answer

Hi  ManojReddy ,

Looks like you have made an error in statement 1 (check highlighted part)

how can x be even and odd simultaneously

x = 2y = z i.e. x MUST be even because y is an Integer and therefore z will also be even

Hence,  SUFFICIENT

I hope it helps!

chetan2u · Answer

1)10^x=2^x*5^x=2^2y*5^z...
equating powers on two sides.. x=2y... suff

2) 3^x*3^5=3^3y*3^3....
so x+2=3y... x can be 4 when y=2 and x will be 7 when y=3... insuff

ans A

DropBear · Answer

I saw this question pop up today when I was studying my MGMAT book. In a recent post by mikemcgarry he encouraged me to not dismiss things and to look deeper to really ensure you understand the mechanics of the problem/question rather than just a surface understanding (something about icebergs comes to mind).

I see in the explanation here (and in the MGMAT book) that we are able to infer that \(4^y\) = \(2^2^y\). I'm certainly not going to argue that this is true as I have tested the theory with a few different numbers, and the rest of the problem makes sense to me. Rather than just dismiss this I would like to grasp the mechanics of why we are able to do this. I hope that if I understand this better I will be able to recognise this more easily in the future and in what cases I can and more importantly cannot make this inferance.

Thanks in advance for the help.

p.s. I know this is probably a very simple question for most of you around here but I guess we all get caught up on simple things from time to time

Bunuel · Answer

4^y=(2^2)^y=2^{2y} . Theory on Exponents and Roots: math-number-theory-88376.html Tips on Exponents and Roots: exponents-and-roots-on-the-gmat-tips-and-hints-174993.html All DS Exponents questions to practice: search.php?search_id=tag&tag_id=39 All PS Exponents questions to practice: search.php?search_id=tag&tag_id=60 All DS roots problems to practice: search.php?search_id=tag&tag_id=49 All PS roots problems to practice: search.php?search_id=tag&tag_id=113 Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

Nunuboy1994 · Answer

Statement 1

There is a pattern

10^2= 5^2 *2^2 
10^3 =5^2 * 2^ 3 
10^4= 2^4 *5^4
10^5= 2^ 5  *5^5

In order to rewrite 10^3 and 10^5 the 2 would have to be written has 4^(3/2)- this cannot be true since it must be an integer so x must be even

suff

Statement 2

Simply rewrite and use algebra

3^ (x+5) = 3^3y + 3

x +5= 3y + 3
 x + 2= 3y

too many possibilities

insuff

A

SPEEDBOATS · Answer

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 searched high and low and could not find this question posted. Please redirect and lock thread if already posted. Also, I couldn't figure out how to format the (x+5) and (y+1) correctly. I read through the entire "Writing Mathematical Formulas on the Forum"...

chetan2u · Answer

Hi...
If you look down below in similar topics, this stands out..
Merging topics

ZMoy · Answer

I know that the OA is A and why that's the case.

However, what if x, y an z were all 0? That'd make x neither positive nor negative. This means that even with the statement 10^x = (4^y)*5^z, the answer could be even or 0.

Doesn't that mean the answer should be E?

WM88 · Answer

I am having a hard time with a simple concept

Statement 1) 
 10^x=(4^y)(5^z) 
Breaking down into prime factorization we get
 (5*2)^x=2^2^y5^z

Which simplifies to 
 x=2y*z

Can somebody explain why people are getting x=2x and x=z

Why isn't it  x=2y*z ?

darkknight10 · Answer

Hi  Bunuel  
From (1) > we can also conclude that x = y = z = 0 is one solution since zero is an integer and nowhere the ques is saying these are distinct numbers.

Is 0 even in GMAT?

What about -2, -4, -6? Even/odd or the concept of even-odd is only for natural numbers?

chetan2u · Answer

0 is even but it is neither positive nor negative.

The above is true for GMAT and all other examinations too.

NEYR0N · Answer

how do we infer that x = 2y and x = z ?

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

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

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