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

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

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.

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.

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


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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?

_______________________
Yes...

Thank you Brother with kudos!

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

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.

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.

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

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