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

Question

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

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

Zarrolou · Answer

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

Bunuel · Answer

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.

Bunuel · Answer

It's 5^z both in the question and in the solution.

HulkSmashGMAT · Answer

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.

TheUltimateWinner · Answer

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...

Bunuel · Answer

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.

TheUltimateWinner · Answer

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

Bunuel · Answer

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.

TheUltimateWinner · Answer

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'

Bunuel · Answer

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

TheUltimateWinner · Answer

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

Bunuel · Answer

_______________________
Yes...

TheUltimateWinner · Answer

Thank you Brother with kudos!

ScottTargetTestPrep · Answer

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

DoISuckAtMath · Answer

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?

Bunuel · Answer

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.

Basshead · Answer

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

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

2^{2x}5^{2x} = 2^y5^z

y - z = 2x - 2x = 0

Sufficient.

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

2^y5^y = 2^{2x}5^{z+x+1}

y = z + x + 1  and  y = 2x . Unable to determine y - z. Insufficient.

Answer is A.

HansJK · Answer

Can we never do this type of equating when the variables are not intergers?

bumpbot · Answer

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

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