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# For integers x, y, and z, x = y^2. What is the value of z?

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For integers x, y, and z, x = y^2. What is the value of z? [#permalink]

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08 Apr 2014, 16:01
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For integers x, y, and z, x = y^2. What is the value of z?

(1) x = z!(z−1)!
(2) 12 < z < 22
[Reveal] Spoiler: OA

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Re: For integers x, y, and z, x = y^2. What is the value of z? [#permalink]

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08 Apr 2014, 23:32
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Mountain14 wrote:
For integers x, y, and z, x=y2. What is the value of z?

1) x=z!(z−1)!
(2) 12<z<22

Given $$x = y^2$$
What is the relevance of y here? The statements only give x and z and we need the value of z. y is introduced here to tell you that x is a perfect square. so all its prime factors have even powers.

1) x=z!(z−1)!

When will x be a perfect square? Whenever z is a perfect square.
We know that z! = z * (z - 1)!

$$x = z * (z - 1)! * (z - 1)! = z * (z - 1)!^2$$
(z -1)! is already squared. So for x to be perfect square, z should also be a perfect square.
z could be 1 or 4 or 9 or 16 or 25 and so on...

(2) 12<z<22
z could be 13 or 14 or 15 and so on...

Using both together, between 12 and 22, there is only one perfect square i.e. 16. So z must be 16.

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For integers x, y, and z, x = y^2. What is the value of z? [#permalink]

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08 Apr 2014, 23:55
Mountain14 wrote:
For integers x, y, and z, x=y2. What is the value of z?

1) x=z!(z−1)!
(2) 12<z<22

Sol:St 1 can be re-written as $$x=y^2= z*(z-1)!^{2}$$

If x=1=y^2 then z!*(z-1)! = 1 or z=1
Now x =y^2 so z*{(z-1)!}^2 will also need to be a perfect square
So if z=25 then we have x=y^2 = 25*(24!)^2 or 25*(24*23*22*21.....1)^2 or 5^2(24*23*22*21....1)^2

So Z can have any value 1,4,9,16,25 and so on and therefore St 1 is not sufficient

from St 2 we know that z is in between 12 and 22 but there is no relation given between x and z or y and z. Therefore not sufficient

Combining both statement we get that z=16.

Ans is C

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For integers x, y, and z, x=y2. What is the value of z? [#permalink]

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08 Jul 2014, 06:05
VeritasPrepKarishma wrote:
Mountain14 wrote:
For integers x, y, and z, x=y2. What is the value of z?

1) x=z!(z−1)!
(2) 12<z<22

Given $$x = y^2$$
What is the relevance of y here? The statements only give x and z and we need the value of z. y is introduced here to tell you that x is a perfect square. so all its prime factors have even powers.

1) x=z!(z−1)!

When will x be a perfect square? Whenever z is a perfect square.
We know that z! = z * (z - 1)!

$$x = z * (z - 1)! * (z - 1)! = z * (z - 1)!^2$$
(z -1)! is already squared. So for x to be perfect square, z should also be a perfect square.
z could be 1 or 4 or 9 or 16 or 25 and so on...

(2) 12<z<22
z could be 13 or 14 or 15 and so on...

Using both together, between 12 and 22, there is only one perfect square i.e. 16. So z must be 16.

Hi Karishma,

I have a query here.

x = y^2. i.e. x is a perfect square. ( but in question where this is telling this is a perfect square. if I will take y =1 then x=y^2 = 1. Now this can not be perfect square bcz 1 is not having any prime factor)perfect square for Here x=z!(z-1)! so if I will take 1 the 1!*0! = 1*1 = 1 we can say 1 is square no. so x=y^2. x=1^2.

I just want to know why ans. can not be A.

Thanks.
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Re: For integers x, y, and z, x = y^2. What is the value of z? [#permalink]

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08 Jul 2014, 06:14
PathFinder007 wrote:
VeritasPrepKarishma wrote:
Mountain14 wrote:
For integers x, y, and z, x=y2. What is the value of z?

1) x=z!(z−1)!
(2) 12<z<22

Given $$x = y^2$$
What is the relevance of y here? The statements only give x and z and we need the value of z. y is introduced here to tell you that x is a perfect square. so all its prime factors have even powers.

1) x=z!(z−1)!

When will x be a perfect square? Whenever z is a perfect square.
We know that z! = z * (z - 1)!

$$x = z * (z - 1)! * (z - 1)! = z * (z - 1)!^2$$
(z -1)! is already squared. So for x to be perfect square, z should also be a perfect square.
z could be 1 or 4 or 9 or 16 or 25 and so on...

(2) 12<z<22
z could be 13 or 14 or 15 and so on...

Using both together, between 12 and 22, there is only one perfect square i.e. 16. So z must be 16.

Hi Karishma,

I have a query here.

x = y^2. i.e. x is a perfect square. ( but in question where this is telling this is a perfect square. if I will take y =1 then x=y^2 = 1. Now this can not be perfect square bcz 1 is not having any prime factor)perfect square for Here x=z!(z-1)! so if I will take 1 the 1!*0! = 1*1 = 1 we can say 1 is square no. so x=y^2. x=1^2.

I just want to know why ans. can not be A.

Thanks.

Let me ask you a question. Is 1 the only possible value of x?
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Re: For integers x, y, and z, x = y^2. What is the value of z? [#permalink]

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08 Jul 2014, 06:56
Bunuel wrote:
PathFinder007 wrote:

Let me ask you a question. Is 1 the only possible value of x?

I got my misttake

Thanks.
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Re: For integers x, y, and z, x = y^2. What is the value of z? [#permalink]

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21 May 2016, 15:33
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Z = ?

Since x = y^2 we know that x is a perfect square.

Statement 1:

x = z!(z-1)!

I think the hardest part of this statement is figuring out a number that satisfies the statement and is a perfect square.

Take z = 5. z!(z-1)! = 5.4.3.2.1(4.3.2.1) which is not a perfect square. But, if we pick a perfect square for z we might be on to something.

z = 4.

4.3.2.1(3.2.1) is a perfect square!!! There, statement 1 is not sufficient as there are lots of different perfect squares!

Statement 2:

Not sufficient as z could be any of those numbers.

Statement 1 & 2:

Is sufficient as the only perfect square between 12 and 22 is 16. Answer C.
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Re: For integers x, y, and z, x = y^2. What is the value of z? [#permalink]

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20 Oct 2016, 06:00
For integers x,y, and z,
x=y^2. What is the value of z?
(1) x=z!(z−1)!
(2) 12<z<22

We know X is a perfect square.
1) x=z!(z−1)! = perfect square

take cases when z=0, the factorial of negative is undefined so Z cannot be zero
when z=1 then x=1!0! = 1- which is a perfect square
when z=2 then x=2!1! = 2 - which is not a perfect square
when z=3 then x=3!2! = 3*2^2 - not a perfect square again
when z=4 then x=4!3! = 4*3^2*2^2=2^2*3^2*2^2 - ok! this is a perfect square.
You should have noticed by now that you will never get a perfect square unless z itself is a perfect square.
so again when Z=9 then x=9!8! =9*8^2*7^2*......*2^2=3^2*8^2*7^2*......*2^2 - here also we have a perfect square
so, Z=1,4,9,16,25,36............
NOT SUFFICIENT.

(2) 12<z<22
so z= 13,14,15,16,17,18,19,20 and 21
Clearly, NOT SUFFICIENT

(1) and (2) Together
Compare the values , You will get a unique value for Z
i.e. Z=16

BOTH SUFFICIENT
Ans C
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For integers x, y, and z, x = y^2. What is the value of z? [#permalink]

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15 Apr 2017, 14:00
VeritasPrepKarishma wrote:
Mountain14 wrote:
For integers x, y, and z, x=y2. What is the value of z?

1) x=z!(z−1)!
(2) 12<z<22

Given $$x = y^2$$
What is the relevance of y here? The statements only give x and z and we need the value of z. y is introduced here to tell you that x is a perfect square. so all its prime factors have even powers.

1) x=z!(z−1)!

When will x be a perfect square? Whenever z is a perfect square.
We know that z! = z * (z - 1)!

$$x = z * (z - 1)! * (z - 1)! = z * (z - 1)!^2$$
(z -1)! is already squared. So for x to be perfect square, z should also be a perfect square.
z could be 1 or 4 or 9 or 16 or 25 and so on...

(2) 12<z<22
z could be 13 or 14 or 15 and so on...

Using both together, between 12 and 22, there is only one perfect square i.e. 16. So z must be 16.

Okay so I think I have a clearer understanding of this problem after reading some of the solutions here:
It is important to pay attention to the fact that the problem reads ""integers" x,y and z etc..) knowing this it becomes clearer to see that x must be a perfect square because
root x = y- which must be an integer; therefore, "X" is a perfect square.

Statement 1:
If X is a perfect square then Z!(Z-!) must be a perfect square. Though there are infinite possibilities for Z- without a restriction this piece of data is insufficient. Insufficient.

Statement 2:
Provides us with a range for Z; however, with no established relationship between x and z we cannot, such as the equation listed in 1, we cannot solve for y. Insufficient.

Statement 1 & Statement 2

If we combine both elements, then we can solve the equation in statement 1 using the criteria in statement 2. Sufficient.
For integers x, y, and z, x = y^2. What is the value of z?   [#permalink] 15 Apr 2017, 14:00
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