GMAT CLUB OLYMPICS: If x 0, is 1/x a prime number ?

Question

If x > 0, is 1/x a prime number ?

(1)  x^x = \frac{1}{\sqrt{2}}

(2) The highest common factor of integers 1/x^2 and 1/x^4 is the square of a prime number

M36-22

Bunuel · Accepted Answer

Official Solution:

If  x > 0 , is  \frac{1}{x}  a prime number ? 
 
(1)  x^x = \frac{1}{\sqrt{2}} 
 
Let's see how we can express  \frac{1}{\sqrt{2}}  so that the base and the exponent are the same (so that to mimic  x^x ).
 
 \frac{1}{\sqrt{2}}=\frac{1}{2^{\frac{1}{2}}}=(\frac{1}{2})^{\frac{1}{2}} ;
 
 So,  x  can be  \frac{1}{2}  
 
We can also express  \frac{1}{\sqrt{2}}  as  \frac{1}{\sqrt{2}}=(\frac{1}{2})^{\frac{1}{2}}=(\frac{1}{2})^{2*\frac{1}{4}}=((\frac{1}{2})^2)^{\frac{1}{4}}=(\frac{1}{4})^{\frac{1}{4}} 
 
 So,  x  can also be  \frac{1}{4}  
 
Not sufficient.
 
(2) The highest common factor of integers  \frac{1}{x^2}  and  \frac{1}{x^4}  is the square of a prime number.
 
Say  \frac{1}{x^2}=k . Then the highest common factor of  k=\frac{1}{x^2}  and  k^2=\frac{1}{x^4}  would obviously be  k . So, we are told that  k=\frac{1}{x^2}  is the square of a prime number:
 
 \frac{1}{x^2}=prime^2 ;
 
 \frac{1}{x}=prime .
 
Sufficient.

Answer: D

yashikaaggarwal · Answer

Statement 1: x^x = 1/√2
Let X = 1/2 
x^x = 1/√2

Let X = 1/4
x^x is still equal to 1/√2
(Not sufficient)

Statement 2:
HCF of 1/x^2 and 1/x^4 = 1/x^2
and 1/x^2 is the square of a prime number
therefore 1/x is a prime number
(Sufficient)

Answer is B

JonShukhrat · Answer

1. Analyzing the first statement we can see that X can be either ½ or ¼. Since ½ power of ½ and ¼ power of ¼ both yield (1/2)^(1/2)

So the answer either can be yes and not. So insufficient.

2. The highest common factor of integers 1/x^2 and 1/x^4 is the square of a prime number

from this choice we can see that 1/x can only be ½, 1/3, 1/5, 1/7, and so on. Statements says that 1/x^2 is an integer. For this x must be a fraction whose reversal is a prime number such as 2, 3 , 5, 7 and so on. only in such case 1/x^2 and 1/x^4 can have the square of a PRIME number as the highest common factor. Sufficient.

So B

Crytiocanalyst · Answer

for statement 1 we can take 2
cases 1: (1/2)^1/2 = 1/(2)^1/2 in this case it's prime 1/x is prime

however case 2 : (1/4) ^1/4 =1/(2)^1/2 in this cae 1/x is not prime

clearly insufficient

for statement 2 
1/x^2 and 1/x^4 has 1/x^2 as common factor and it explicitly states it's a prime factor squared which implies 1/x is prime clearly sufficient

Therefore IMO B

GMATinsight · Answer

Answer: Option B

Step-by-Step  Video solution by  GMATinsight

https://www.youtube.com/watch?v=uq1sLImO1qU

raju1212 · Answer

Can someone explain how 1/x ever be a prime number. I thought prime numbers need to be intergers and hence a fraction cannot be prime.

chetan2u · Answer

If x is a fraction, then 1/x can be an integer. 
For example x=1/2 means 1/x=2

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GMAT CLUB OLYMPICS: If x > 0, is 1/x a prime number ?

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