If x, y, and z are integers greater than 1, and (3^27)(35^10)(z) = (5^8)(7^10)(9^14)(x^y), then what is the value of x?
(1) z is prime
(2) x is prime
OA is D
I simplified the original equation to [spoiler]25z=3x^y[/spoiler] but couldnt deduce anything based on the statements. how can i logically figure this out?
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GMATGuruNY
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(3²�)(35¹�)(z) = (5�)(7¹�)(9¹�)(x^y)hutch27 wrote:If x, y, and z are integers greater than 1, and (3^27)(35^10)(z) = (5^8)(7^10)(9^14)(x^y), then what is the value of x?
(1) z is prime
(2) x is prime
OA is D
I simplified the original equation to [spoiler]25z=3x^y[/spoiler] but couldnt deduce anything based on the statements. how can i logically figure this out?
(3²�)(7¹�5¹�)(z) = (5�)(7¹�)(3²)¹�(x^y)
(3²�)(7¹�5¹�)(z) = (5�)(7¹�)(3²�)(x^y)
(5²)(z) = (3)(x^y)
z = (3) * (x^y)/5².
The equation above implies that z is a multiple of 3 and that x^y is a multiple of 5².
Statement 1: z is prime
Since z is prime and a multiple of 3, z=3.
Thus, (x^y)/5² = 1, implying that x=5 and y=2.
SUFFICIENT.
Statement 2: x is prime
Since x^y is a multiple of 5² and x is prime, x=5 and y≥2.
SUFFICIENT.
The correct answer is D.
Last edited by GMATGuruNY on Tue Oct 08, 2013 5:01 am, edited 1 time in total.
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ceilidh.erickson
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When you have variables as exponents (and all the variables are integers), you just need to ask yourself - what would allow one side to match the other?
You simplified the equation to 25z=3x^y, which is exactly right. Now ask - what would allow both sides to be equal?
- z has to have at least a 3 in it, because there's no 3 in 25
- x has to have at least a 5 in it, because it needs to account for the 25
- we don't know yet if z and x have other factors in them - 2's or 7's, etc. If x has other stuff in it, then z will have to match that stuff.
So, in order to figure out what x is, we'd need to know know what restrictions there are on the possibilities.
1) z is prime
Since we know that z has to have at least a 3, then that must be the prime - z doesn't have anything else in it. And if z is 3, then 25*3 = x^y, so x must be 5 and y must be 2. Sufficient.
2) x is prime
We know that x has to have at least a 5, so x must equal 5. Sufficient.
You simplified the equation to 25z=3x^y, which is exactly right. Now ask - what would allow both sides to be equal?
- z has to have at least a 3 in it, because there's no 3 in 25
- x has to have at least a 5 in it, because it needs to account for the 25
- we don't know yet if z and x have other factors in them - 2's or 7's, etc. If x has other stuff in it, then z will have to match that stuff.
So, in order to figure out what x is, we'd need to know know what restrictions there are on the possibilities.
1) z is prime
Since we know that z has to have at least a 3, then that must be the prime - z doesn't have anything else in it. And if z is 3, then 25*3 = x^y, so x must be 5 and y must be 2. Sufficient.
2) x is prime
We know that x has to have at least a 5, so x must equal 5. Sufficient.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
