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If x, y, and z are integers greater than 1, and [#permalink]
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 21 Feb 2010, 14:20
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If x, y, and z are integers greater than 1, and \(3^{27}*5^{10}*z = 5^8*9^{14}*x^y\), then what is the value of x? (1) y is prime. (2) x is prime.
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Last edited by Bunuel on 08 Oct 2012, 04:18, edited 2 times in total.
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Re: If x, y, and z are integers [#permalink]
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22 Feb 2010, 11:19
(327)(510)(z) = (58)(914)(xy)
3 * 109 * 3 * 17 * 2 * 5 * z = 2 * 29 * 2 * 457 * x * y (all the numbers are prime)
x = (3^2 * 5 * 17 * 109 * z) / (29 * 2 * 457 * y) (cancelling 2 on numerator and denominator)
1) y is prime
Not sufficient
y = 3, z = 29*2*457 * n (where n is an intereger ) and values of x could vary depending on n or depending on y(3,5,109, 457 etc)
(2) x is prime
Not sufficient
y = 9 * 5 * 17,z = 29*2*457 , x = 109
y = 9 * 5 * 109,z = 29*2*457 , x = 17
combining .. there is no value of x that will satify the equation as if y is prime, then y could be any of the values of 3,5, 17, 109 or a different number that is a factor of z. and if z is an interger, x cannot be prime.
E ( not sure if E is the correct answer as there is no valid value of x that satisfies the conditions)
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Re: If x, y, and z are integers [#permalink]
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24 Feb 2010, 16:29
IMO C Reduced equation becomes \(\frac{25*z}{3^{15}*x}\) = y now if y is prime z can be 3^15 and x = 5 or z = 2* 3^15 and x = 10 thus not sufficient now if x is prime same explanation above. now take both x and y prime, in this case x can be only 5 and z = 3^15 thus it should be C
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Re: If x, y, and z are integers [#permalink]
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24 Feb 2010, 16:36
Here's the solution for the re-posted problem  If x, y, and z are integers greater than 1, and 3^27 * 5^10 * z =5^8 * 9^14 * x * y , then what is the value of x? 3^27 * 5^10 * z =5^8 * 9^14 * x * y 3^27 * 5^ 10 * z = 5^ 8 * 3^28 * x * y 5^2 * z = 3 * x * y x = (5^2 * z) / 3y (1) y is prime y can be a factor of z, and z can be a factor of y and there could be a lot of possibilities of x y = 3, z = 27, x = 75 y = 5, z = 3, x = 5 Not sufficient (2) x is prime z has to be a multiple of 3 and y has to be a multiple of 5 z= 3*a (where a is prime not equal to 5), y = 25, then x=a(many values for a) z=3, y = 5, x=5 Not sufficient combining .. x and y both are prime .. and as 5^2 is in the numerator, y has to be 5 and z has be to 3 or else x cannot be prime. y=5, z=3, x=5 C
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Re: If x, y, and z are integers [#permalink]
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24 Feb 2010, 16:45
After simplifying the given question, we end up with 5*5*z/(3*y) = x. Stmt 1 - y can take a whole lot of prime numbers from 2 to infinity (not the car!!!) and until we know z, we can't comment on value of x. Stmt 2 - x itself can take a whole lot of prime numbers from 2 to infinity (again, not the car!!!) and we would have to struggle to get appropriate values for z and y, and there can very many that match the criteria. Combining both stmts - as long as z is some composite, that can be expressed as a multiple of 3, x (any prime number choice) and y (prime number choice), it is good, and there are whole lot of possibilities for x as well. I am not very sure, how do we get to B as the OA. In my opinion, it must be E. Math experts and Math God Bunuel might wish to throw some light!!!!
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Re: If x, y, and z are integers [#permalink]
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24 Feb 2010, 17:19
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Re: If x, y, and z are integers [#permalink]
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I had this question in the test today too. I went with E but QA is B. I didnt understand the explanation. I think QA is wrong here, 2 is a prime too.
---
The best way to answer this question is to use the rules of exponents to simplify the question stem, then analyze each statement based on the simplified equation.
(327)(510)(z) = (58)(914)(xy) Simplify the 914
(327)(510)(z) = (58)(328)(xy) Divide both sides by common terms 58, 327
(52)(z) = 3xy
(1) INSUFFICIENT: Analyzing the simplified equation above, we can conclude that z must have a factor of 3 to balance the 3 on the right side of the equation. Similarly, x must have at least one factor of 5. Statement (1) says that y is prime, which does no tell us how many fives are contained in x and z.
For example, it is possible that x = 5, y = 2, and z = 3:
52 · 3 = 3 · 52
It is also possible that x = 25, y = 2, and z = 75:
52 · 75 = 3 · 252
52 · 52 · 3 = 3 · 252
(2) SUFFICIENT: Analyzing the simplified equation above, we can conclude that x must have a factor of 5 to balance out the 52 on the left side. Since statement (2) says that x is prime, x cannot have any other factors, so x = 5. Therefore statement (2) is sufficient.
The correct answer is B.
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Re: If x, y, and z are integers [#permalink]
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what if y=25 , z=6 and x=2 ? if y =5, x=5 and z=3 ? for statement 2 both holds true..so B alone is not sufficient. either OA is wrong or Question is wrong.
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Re: If x, y, and z are integers [#permalink]
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02 Aug 2010, 04:50
The answer has to be C..
3^27 * 5^10 * z =5^8 * 9^14 * x * y
3^27 * 5^ 10 * z = 5^ 8 * 3^28 * x * y
5^2 * z = 3 * x * y
x = (5^2 * z) / 3y
after this..either of the statements ie..
1) X is prime
y and z can take a whole lot of values..
2) y is prime.
x and z can again take many values..prime or not prime..
hence only when we know that both x and y are prime
can we reach the answer that z=3, x=5,y=5
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Re: If x, y, and z are integers [#permalink]
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27 Jun 2012, 09:36
Bunuel wrote: jeeteshsingh wrote: If x, y, and z are integers greater than 1, and \(3^{27}*5^{10}*z = 5^8*9^{14}*x*y\), then what is the value of x? (1) y is prime (2) x is prime SOURCE: Manhattan tests Please explain in detail If the answer is really B, then I think question should be: \(3^{27}*5^{10}*z = 5^8*9^{14}*x^y\). If I'm right, then it's B indeed. In it's current form the answer is C as explained. Bunuel, can you please explain the answer for choice B
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Re: If x, y, and z are integers [#permalink]
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riteshgupta wrote: Bunuel wrote: jeeteshsingh wrote: If x, y, and z are integers greater than 1, and \(3^{27}*5^{10}*z = 5^8*9^{14}*x*y\), then what is the value of x? (1) y is prime (2) x is prime SOURCE: Manhattan tests Please explain in detail If the answer is really B, then I think question should be: \(3^{27}*5^{10}*z = 5^8*9^{14}*x^y\). If I'm right, then it's B indeed. In it's current form the answer is C as explained. Bunuel, can you please explain the answer for choice B You mean what would be the solution if it were x^y instead of xy? If x, y, and z are integers greater than 1, and \(3^{27}*5^{10}*z = 5^8*9^{14}*x^y\), then what is the value of x?\(3^{27}*5^{10}*z = 5^8*9^{14}*x^y\) --> \(3^{27}*5^{2}*z =3^{28}*x^y\) --> \(5^{2}*z = 3*x^y\) --> \(\frac{x^y}{z}=\frac{5^2}{3}\), so \(x^y\) is a multiple of 25 and \(z\) is a multiple of 3. (1) y is prime. We can have that \(x=5\), \(y=2=prime\) and \(z=3\) OR \(x=10\), \(y=2=prime\) and \(z=12\)... Not sufficient. (2) x is prime. Since \(x^y\) is a multiple of \(5^2\) and \(x\) is a prime, then \(x=5\). Sufficient. Answer: B. Hope it's clear.
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Re: If x, y, and z are integers greater than 1, and [#permalink]
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17 Jul 2012, 08:20
think the answer is B even as written?
Formula simplifies to 25z/3y = x or 25z/3x=y
1: y is prime; since 25z/3y yields an integer, does it not hold that 25/y = integer since 3 is not a factor of 25. If y is prime, it has to be 5 since 5^2 are the prime factors of 25. However, in order to know X, we need to know what z might be, and Z could be any number with 3 as a prime factor. Not sufficient.
2: x is prime; since 25z/3x yields an integer, 25/x is also an integer since 3 is not a factor of 25. Holding the same logic as in 1; x must be 5. Sufficient.
Please let me know if I am thinking about this wrong
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Re: If x, y, and z are integers greater than 1, and [#permalink]
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17 Jul 2012, 16:41
JohnGalt44 wrote: think the answer is B even as written?
Formula simplifies to 25z/3y = x or 25z/3x=y
1: y is prime; since 25z/3y yields an integer, does it not hold that 25/y = integer since 3 is not a factor of 25. If y is prime, it has to be 5 since 5^2 are the prime factors of 25. However, in order to know X, we need to know what z might be, and Z could be any number with 3 as a prime factor. Not sufficient.
2: x is prime; since 25z/3x yields an integer, 25/x is also an integer since 3 is not a factor of 25. Holding the same logic as in 1; x must be 5. Sufficient.
Please let me know if I am thinking about this wrong Yes you are thinking about it wrongly. Why does 25/x have to be an integer? 3 is not a factor of 25, but it can be a factor of z, the other term in the numerator. Hence, if you let y=5^2, then z=3x, and x=any prime you want it to be. Not sufficient However taking (1)+(2), both x and y have to be prime and since you need to make 25 using 3*x*y, only way to do it is using x=y=5 and z=3. C. Please give kudos if you like. Also, can OP please change answer in spoiler to the correct one?
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Re: If x, y, and z are integers greater than 1, and [#permalink]
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jeeteshsingh wrote: If x, y, and z are integers greater than 1, and \(3^{27}*5^{10}*z = 5^8*9^{14}*x^y\), then what is the value of x?
(1) y is prime.
(2) x is prime. \(3^{27}*5^{10}*z = 5^8*9^{14}*x^y\) \(3*9^{13}*5^{10}*z = 5^8*9^{14}*x^y\) \(5^2*z = 3*x^y\) \(x^y = \frac{5^2*z}{3}\) It's obvious that z has a factor of 3 to cancel out the denominator and x has 5 as a factor... 1. y is prime Let y = 3: \(5^3 = \frac{5^2*(5*3)}{3}\) Thus, x=5 Let y = 3 and x contain 3: \(5^{3}*3^3 = \frac{5^2*(5*3^4)}{3}\) Thus, x=15 INSUFFICIENT. 2. x is prime.. Well it's obvious that x has 5 as a factor.. If it's prime then x = 5*1 = 5 SUFFICIENT. Answer: B
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Re: If x,yand z are integers greater than 1,and [#permalink]
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24 Feb 2013, 06:18
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Re: If x, y, and z are integers [#permalink]
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18 Jul 2013, 15:05
Bunuel wrote:
You mean what would be the solution if it were x^y instead of xy?
If x, y, and z are integers greater than 1, and \(3^{27}*5^{10}*z = 5^8*9^{14}*x^y\), then what is the value of x?
\(3^{27}*5^{10}*z = 5^8*9^{14}*x^y\) --> \(3^{27}*5^{2}*z =3^{28}*x^y\) --> \(5^{2}*z = 3*x^y\) --> \(\frac{x^y}{z}=\frac{5^2}{3}\), so \(x^y\) is a multiple of 25 and \(z\) is a multiple of 3.
(1) y is prime. We can have that \(x=5\), \(y=2=prime\) and \(z=3\) OR \(x=10\), \(y=2=prime\) and \(z=12\)... Not sufficient.
(2) x is prime. Since \(x^y\) is a multiple of \(5^2\) and \(x\) is a prime, then \(x=5\). Sufficient.
Answer: B.
Hope it's clear.
Bunuel, if the statement (1) were- Z (instead of Y) is prime then the answer should be 'D'? Thanks!
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Re: If x, y, and z are integers [#permalink]
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18 Jul 2013, 15:17
mneeti wrote: Bunuel wrote:
You mean what would be the solution if it were x^y instead of xy?
If x, y, and z are integers greater than 1, and \(3^{27}*5^{10}*z = 5^8*9^{14}*x^y\), then what is the value of x?
\(3^{27}*5^{10}*z = 5^8*9^{14}*x^y\) --> \(3^{27}*5^{2}*z =3^{28}*x^y\) --> \(5^{2}*z = 3*x^y\) --> \(\frac{x^y}{z}=\frac{5^2}{3}\), so \(x^y\) is a multiple of 25 and \(z\) is a multiple of 3.
(1) y is prime. We can have that \(x=5\), \(y=2=prime\) and \(z=3\) OR \(x=10\), \(y=2=prime\) and \(z=12\)... Not sufficient.
(2) x is prime. Since \(x^y\) is a multiple of \(5^2\) and \(x\) is a prime, then \(x=5\). Sufficient.
Answer: B.
Hope it's clear.
Bunuel, if the statement (1) were- Z (instead of Y) is prime then the answer should be 'D'? Thanks! Yes, that's correct.
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If x, y and z are integers [#permalink]
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12 Oct 2013, 09:41
If x, y, and z are integers greater than 1, and (327)(3510)(z) = (58)(710)(914)(xy), then what is the value of x?
(1) z is prime
(2) x is prime
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Re: If x, y and z are integers [#permalink]
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Re: If x, y, and z are integers [#permalink]
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15 Oct 2013, 21:44
Bunuel wrote: If x, y, and z are integers greater than 1, and \(3^{27}*5^{10}*z = 5^8*9^{14}*x*y\), then what is the value of x? (1) y is prime (2) x is prime SOURCE: Manhattan tests Please explain in detail It comes out this way- 5 X 5 X Z = 3 X \(x^y\) How is B correct?
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Re: If x, y, and z are integers
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