If z is a positive integer is root(Z) an integer ?

so it means that:

case 1: if x = 9 (perfect sq) and z = cannot be any integer as it will not satisfy the equation sqrt xz = i... it has to be a perfect sq it self, such as 4... sqrt 9*4 = 6

Case 2: but if it x - not a perfect square say 3, then z will have to be 3 so satisfy the equation sqrt xz = Integer, but in that case sqrt z will not be an integer..

Case 3: if x is a reduced fraction: 5/2 then z will have to be 2 or 8 to satisfy the equation sqrt xz = Integer and even in this case sqrt z is not an integer

so the fact that \sqrt{z}is an integer depends on value of x.. therefore, Insufficient.

I hope i have finally understood.

st 2... in this x= z3 means nothing as z= cuberoot x... x can be 27 in this case z= 3 but \sqrt{z}= irrational number
or x can be 64.. making z = 4 and \sqrt{z} and integer... Insufficient again

together (st 1 & st2): \sqrt{xz} = \sqrt{z^4} = z^2 = Integer.... but \sqrt{z} can be an integer or irrational number

i think i have got it now. Am i?

(1) Not sufficient. Eg. Z=25, X=25 ; Z=3, X=3
(2) X=Z^3. No relation to Z. Not sufficient

(1)+(2) Not sufficient. Eg. Z=25, X=25^3 ; Z=3, X=3^3

Answer is (E)

HI Bunnel

when you say \sqrt{xz}=integer and we already know that z is an Integer... I*I = I?
i read in your other posts about irrational numbers: z cannot be an irrational number here as it is already given that it is a integer... therefore, shouldnt x also be an integer..
basically how would knowing anything about x help?

The stem says that \(z=integer\) and (1) says \(\sqrt{xz}=integer\). Is it necessary for x to be integer?

No. For example, consider \(z=2\) and \(x=\frac{9}{2}\) --> \(\sqrt{xz}=3=integer\).

Even if we knew that x were an integer the first statement still would not be sufficient: consider \(z=2\) (\(\sqrt{z}=\sqrt{2}\neq{integer}\)) and \(x=2\) --> \(\sqrt{xz}=2=integer\).

Hope it's clear.

hi

but in both examples z = sqrt 2 which is not an integer.... i am confused regarding the fact that y who we need to know about x...
for example: if z=2 and x = 2, sqrt xz= Integer but sqrt z is not and integer.
if z = 4 and x = 9/4, still sqrt xz = 3, Integer but sqrt z is an integer... so how is knowing the value of x even relevant for the qs?

Are you asking what should we know about x, so that the first statement would be sufficient? If for example, we were told that x is a perfect square, then the first statement would be sufficient to say that \(\sqrt{z}=integer\).

Easy E
from statement 1
x can be 8 and z can be 2
or x can be 9 and z can 4
hence insufficient
from statement 2
x=z^3
it can be anything since the nature of x is unknown
combining 1 and 2
we get \sqrt{z^4} = z^2 again insufficient z can be 2 or 4
hence answer is E

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