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Intern  Joined: 26 Jun 2012
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GMAT 1: 570 Q31 V39 GMAT 2: 710 Q43 V44 If x, y and z are integers, what is y – z?  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 68% (02:21) correct 32% (02:15) wrong based on 349 sessions

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If x, y and z are integers, what is y – z?

(1) $$100^x = 2^y5^z$$

(2) $$10^y = 20^x5^{z+1}$$
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Re: If x, y and z are integers, what is y – z?  [#permalink]

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1
If x, y and z are integers, what is y – z?

(1) $$100^x = 2^y5^z$$
$$2^{2x}5^{2x}=2^y5^z$$ so $$y-z=2x-2x=0$$.
Sufficient

(2) $$10^y = 20^x5^{z+1}$$
$$2^y5^y=2^{2x}5^x5^{z+1}$$ so $$y=2x$$ and $$y=x+z+1$$. We cannot determine y-z.
Consider y=4,x=2 and z=1 so y-z=3; or y=8,x=4 and y=3 so y-z=5.
Not sufficient

A
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If x, y and z are integers, what is y – z?  [#permalink]

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If x, y and z are integers, what is y – z?

(1) $$100^x = 2^y5^z$$ --> $$2^{2x}5^{2x}=2^y5^z$$ --> equate the exponents: $$2x=y$$ and $$2x=z$$ --> thus $$2x-2x=y-z=0$$. Sufficient.

(2) $$10^y = 20^x5^{z+1}$$ --> $$2^y5^y=2^{2x}*5^{x+z+1}$$ --> $$y=2x$$ and $$y=x+z+1$$. We cannot get the value of y-z from this. Not sufficient,

Answer: A.

Hope it's clear.
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Re: If x, y and z are integers, what is y – z?  [#permalink]

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ashutoshyadav1707 wrote:
Bunuel wrote:
vishalrastogi wrote:
I could not get the explanation here, can anybody explain this, please ?

If x, y and z are integers, what is y – z?

(1) $$100^x = 2^y5^z$$ --> $$2^{2x}5^{2x}=2^y5^z$$ --> equate the exponents: $$2x=y$$ and $$2x=z$$ --> thus $$2x-2x=y-z=0$$. Sufficient.

(2) $$10^y = 20^x5^{z+1}$$ --> $$2^y5^y=2^{2x}*5^{x+z+1}$$ --> $$y=2x$$ and $$y=x+z+1$$. We cannot get the value of y-z from this. Not sufficient,

Answer: A.

Hope it's clear.

How can 1 be sufficient??? In the given statement, its 5 raise to the power 2. And the solution you have provided considers it as 5 raise to the power z.

It's 5^z both in the question and in the solution.
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Re: If x, y and z are integers, what is y – z?  [#permalink]

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1
I think what's confusing some folks is the second equation is giving y = x+z+1 bringing z to the left side. y-z = x+1.
This still doesn't give a value for y-z. Question is asking for a value for y-z and not if you can deduce an expression for y-z. I made this silly mistake once in the heat of the moment so sharing it here.
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GMAT 1: 720 Q49 V40 Re: If x, y and z are integers, what is y – z?  [#permalink]

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Bunuel wrote:
vishalrastogi wrote:
I could not get the explanation here, can anybody explain this, please ?

If x, y and z are integers, what is y – z?

(1) $$100^x = 2^y5^z$$ --> $$2^{2x}5^{2x}=2^y5^z$$ --> equate the exponents: $$2x=y$$ and $$2x=z$$ --> thus $$2x-2x=y-z=0$$. Sufficient.

Answer: A.

Hope it's clear.

If y-z=0, then y=z.
If i put the value of y=z, how can we legitimate the statement 1?
statement 1:
$$100^x = 2^y5^z$$
$$2^{2x}5^{2x}=2^z5^z$$
To legitimate the statement 1 we still need the value of x and z. But, they are still unknown here. How can you make known it for all?
Then, how can we conclude it? Bunuel
Thank you...
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Math Expert V
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Re: If x, y and z are integers, what is y – z?  [#permalink]

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1
iMyself wrote:
Bunuel wrote:
vishalrastogi wrote:
I could not get the explanation here, can anybody explain this, please ?

If x, y and z are integers, what is y – z?

(1) $$100^x = 2^y5^z$$ --> $$2^{2x}5^{2x}=2^y5^z$$ --> equate the exponents: $$2x=y$$ and $$2x=z$$ --> thus $$2x-2x=y-z=0$$. Sufficient.

Answer: A.

Hope it's clear.

If y-z=0, then y=z.
If i put the value of y=z, how can we legitimate the statement 1?
statement 1:
$$100^x = 2^y5^z$$
$$2^{2x}5^{2x}=2^z5^z$$
To legitimate the statement 1 we still need the value of x and z. But, they are still unknown here. How can you make known it for all?
Then, how can we conclude it? Bunuel
Thank you...

The question asks the value of y - z, not the individual value of x, y, and z. From the solution we got that y - z = 0.
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GMAT 1: 720 Q49 V40 Re: If x, y and z are integers, what is y – z?  [#permalink]

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But how did you get the value of y-z , i did not get from your explanation actually. Thank you...

Sent from my iPhone using GMAT Club Forum mobile app
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Math Expert V
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Re: If x, y and z are integers, what is y – z?  [#permalink]

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1
iMyself wrote:
But how did you get the value of y-z , i did not get from your explanation actually. Thank you...

Sent from my iPhone using GMAT Club Forum mobile app

x, y, and z are given to be integers.
We have $$2^{2x}5^{2x}=2^y5^z$$ --> equate the exponents of 2 and 5: $$2x=y$$ and $$2x=z$$. Thus 2x = y = z.
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GMAT 1: 720 Q49 V40 Re: If x, y and z are integers, what is y – z?  [#permalink]

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Bunuel wrote:
iMyself wrote:
But how did you get the value of y-z , i did not get from your explanation actually. Thank you...

Sent from my iPhone using GMAT Club Forum mobile app

x, y, and z are given to be integers.
We have $$2^{2x}5^{2x}=2^y5^z$$ --> equate the exponents of 2 and 5: $$2x=y$$ and $$2x=z$$. Thus 2x = y = z.

This is the first time i learn that i can equate the exponent after having multiple variables on both side. I, normally, equate the exponent when i have only one part in the right hand side and the other one in the left hand side. like below...
2^{2x}=2^y
--> 2x=y
it is ok.
But when it is something like below then it is the first time i learn.
$$2^{2x}5^{2x}=2^y5^z$$
$$2x=y$$ and $$2x=z$$.
Anyway, many many thanks with 'kudos'
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Henry Wadsworth Longfellow

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Math Expert V
Joined: 02 Sep 2009
Posts: 55150
Re: If x, y and z are integers, what is y – z?  [#permalink]

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1
iMyself wrote:
Bunuel wrote:
iMyself wrote:
But how did you get the value of y-z , i did not get from your explanation actually. Thank you...

Sent from my iPhone using GMAT Club Forum mobile app

x, y, and z are given to be integers.
We have $$2^{2x}5^{2x}=2^y5^z$$ --> equate the exponents of 2 and 5: $$2x=y$$ and $$2x=z$$. Thus 2x = y = z.

This is the first time i learn that i can equate the exponent after having multiple variables on both side. I, normally, equate the exponent when i have only one part in the right hand side and the other one in the left hand side. like below...
2^{2x}=2^y
--> 2x=y
it is ok.
But when it is something like below then it is the first time i learn.
$$2^{2x}5^{2x}=2^y5^z$$
$$2x=y$$ and $$2x=z$$.
Anyway, many many thanks with 'kudos'

We can only do this here because we know that x, y, and z are integers.
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GMAT 1: 720 Q49 V40 Re: If x, y and z are integers, what is y – z?  [#permalink]

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Bunuel wrote:
iMyself wrote:
Bunuel wrote:

x, y, and z are given to be integers.
We have $$2^{2x}5^{2x}=2^y5^z$$ --> equate the exponents of 2 and 5: $$2x=y$$ and $$2x=z$$. Thus 2x = y = z.

This is the first time i learn that i can equate the exponent after having multiple variables on both side. I, normally, equate the exponent when i have only one part in the right hand side and the other one in the left hand side. like below...
2^{2x}=2^y
--> 2x=y
it is ok.
But when it is something like below then it is the first time i learn.
$$2^{2x}5^{2x}=2^y5^z$$
$$2x=y$$ and $$2x=z$$.
Anyway, many many thanks with 'kudos'

We can only do this here because we know that x, y, and z are integers.

That means: we can't equate this type of things if the variable is NOT integer, right Bunuel?
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Re: If x, y and z are integers, what is y – z?  [#permalink]

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iMyself wrote:
That means: we can't equate this type of things if the variable is NOT integer, right Bunuel?

_______________________
Yes...
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GMAT 1: 720 Q49 V40 Re: If x, y and z are integers, what is y – z?  [#permalink]

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Bunuel wrote:
iMyself wrote:
That means: we can't equate this type of things if the variable is NOT integer, right Bunuel?

_______________________
Yes...

Thank you Brother with kudos!
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Re: If x, y and z are integers, what is y – z?  [#permalink]

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kingflo wrote:
If x, y and z are integers, what is y – z?

(1) $$100^x = 2^y5^z$$

(2) $$10^y = 20^x5^{z+1}$$

We need to determine the value of y – z.

Statement One Alone:

100^x = 2^y * 5^z

Notice that 100^x = (2^2 * 5^2)^x = 2^(2x) * 5^(2x). Equate this with 2^y * 5^z and we have:

2^(2x) * 5^(2x) = 2^y * 5^z

Therefore, 2^(2x) = 2^y and 5^(2x) = 5^z.

Thus, 2x = y and 2x = z. Therefore, y - z = 2x - 2x = 0.

Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

10^y = 20^x * 5^(z+1)

Since 20^x = (2^2 * 5)^x = 2^(2x) * 5^x, that means 20^x * 5^(z+1) = 2^(2x) * 5^x * 5^(z+1) = 2^(2x) * 5^(x+z+1). Notice that 10^y = (2 * 5)^y = 2^y * 5^y, so we have:

2^y * 5^y = 2^(2x) * 5^(x+z+1)

Therefore, 2^y = 2^(2x) and 5^y = 5^(x+z+1).

Thus, y = 2x and y = x + z + 1. From the second equation, we have y - z = x + 1. However, since we do not know the value of x, we cannot determine the value of y - z. Statement two alone is not sufficient to answer the question.

Answer: A
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Re: If x, y and z are integers, what is y – z?  [#permalink]

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kingflo wrote:
If x, y and z are integers, what is y – z?

(1) $$100^x = 2^y5^z$$

(2) $$10^y = 20^x5^{z+1}$$

Agree to the explanations given. However, if x=y=z=0, then the answer must be E. Neither the initial question task nor each of the two conditions stipulate that x can't equal y and z or 0. Why am I not correct?
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Re: If x, y and z are integers, what is y – z?  [#permalink]

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DoISuckAtMath wrote:
kingflo wrote:
If x, y and z are integers, what is y – z?

(1) $$100^x = 2^y5^z$$

(2) $$10^y = 20^x5^{z+1}$$

Agree to the explanations given. However, if x=y=z=0, then the answer must be E. Neither the initial question task nor each of the two conditions stipulate that x can't equal y and z or 0. Why am I not correct?

If answer is A, then it's A no matter which (acceptable) values you substitute.

The question asks to find the value of y – z. From (1) we goth that y - z = 0. If x = y = z = 0, then y - z is still 0.
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Re: If x, y and z are integers, what is y – z?  [#permalink]

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_________________ Re: If x, y and z are integers, what is y – z?   [#permalink] 11 Jan 2019, 08:47
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