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

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Math Expert
Joined: 02 Sep 2009
Posts: 45459
If, x, y, and z are integers, is x even? [#permalink]

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23 Jun 2015, 02:47
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If, x, y, and z are integers, is x even?

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

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

Kudos for a correct solution.

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Re: If, x, y, and z are integers, is x even? [#permalink]

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23 Jun 2015, 05:22
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
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Re: If, x, y, and z are integers, is x even? [#permalink]

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23 Jun 2015, 10:36
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Bunuel wrote:
If, x, y, and z are integers, is x even?

(1) 10^x = 4^y*5^z
(2) 3^(x + 5) = 27^(y + 1)

Kudos for a correct solution.

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

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Re: If, x, y, and z are integers, is x even? [#permalink]

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23 Jun 2015, 10:38
ManojReddy wrote:
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

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!
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Math Expert
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Posts: 5784
Re: If, x, y, and z are integers, is x even? [#permalink]

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23 Jun 2015, 10:46
Bunuel wrote:
If, x, y, and z are integers, is x even?

(1) 10^x = 4^y*5^z
(2) 3^(x + 5) = 27^(y + 1)

Kudos for a correct solution.

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
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Re: If, x, y, and z are integers, is x even? [#permalink]

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07 Aug 2015, 14:58
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
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Posts: 45459
Re: If, x, y, and z are integers, is x even? [#permalink]

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16 Aug 2015, 11:58
DropBear wrote:
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

$$4^y=(2^2)^y=2^{2y}$$.

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

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Re: If, x, y, and z are integers, is x even? [#permalink]

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25 Sep 2017, 18:13
Bunuel wrote:
If, x, y, and z are integers, is x even?

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

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

Kudos for a correct solution.

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
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Joined: 19 Jun 2017
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If x, y, and z are integers, is x even? [#permalink]

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23 Oct 2017, 19:15
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"...
Math Expert
Joined: 02 Aug 2009
Posts: 5784
Re: If x, y, and z are integers, is x even? [#permalink]

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23 Oct 2017, 19:30
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SPEEDBOATS wrote:
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"...

Hi...
If you look down below in similar topics, this stands out..
Merging topics
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Re: If x, y, and z are integers, is x even?   [#permalink] 23 Oct 2017, 19:30
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# If, x, y, and z are integers, is x even?

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