12 Days of Christmas GMAT Competition - Day 6: If x is a positive

Question

12 Days of Christmas 2024 - 2025 Competition with $40,000 of Prizes

If x is a positive number,  f(x) = \sqrt{x} , and  f(f(x))  is the reciprocal of a prime number, which of the following could be the value of x?

A. 0.25
B. 0.04
C. 0.016
D. 0.008
E. 0.0016

Bunuel · Accepted Answer

GMAT Club's Official Explanation:

Given  f(x) = \sqrt{x} , then:

f(\sqrt{x}) =

=\sqrt{\sqrt{x}} =

=\sqrt {x}

Since  \sqrt {x} = \frac{1}{prime} , taking this to the fourth power gives  x = \frac{1}{prime^4} .

Let's check the options:

A.  0.25 = \frac{1}{4} = \frac{1}{2^2} . Discard

B.  0.04 = \frac{4}{100} = \frac{1}{25} = \frac{1}{5^2} . Discard.

C.  0.016 = \frac{16}{1,000} = \frac{2}{125} = \frac{2}{5^3} . Discard.

D.  0.008 = \frac{8}{1,000} = \frac{1}{125} = \frac{1}{5^3} . Discard.

E.  0.0016 = \frac{16}{10,000} = \frac{1}{625} = \frac{1}{5^4} . Bingo.

Answer: E.

Kinshook · Answer

If x is a positive number,  f(x) = \sqrt{x} , and  f(f(x))  is the reciprocal of a prime number, which of the following could be the value of x?

f(x) = \sqrt{x} = x^{.5} 
 f(f(x)) = f(x^{.5}) = (x^{.5})^{.5} = x^{.25}

Case 1: x^{.25} = 1/2 ; x = (1/2)^4 = 1/16 = 0.0625
Case 2: x^{.25} = 1/3; x = (1/3)^4 = 1/81 = 0.0123456789
Case 3: x^{.25} = 1/5; x = (1/5)^4 = 1/625 = 0.0016

IMO E

siddhantvarma · Answer

f(x) = √ x 
f(f(x)) = √ √ x =>  x^1/4

f(f(x)) = 1/p where p is prime
  x^1/4  = 1/p =>  p^4  = x

We basically need to check for each option, that when you take it's reciprocal and take it's 4th root, do you get a prime number or not.

0.25: 25/100 => 100/24 = 4 (4th root of 4 is not an integer, eliminate)
 0.04: 4/100 => 100/4 = 25 (4th root of 25 is not an integer, eliminate)
 0.016: 16/100 => 100/16 (4th root of 100 is not an integer, eliminate)
 0.008: 8/1000 => 1000/8 (4th root of 1000 is not an integer eliminate)
 0.0016: 16/10000 => 10000/16 = 10/2 = 5 (4th root is 5 which is a prime number)

Answer: E. 0.0016

HarshaBujji · Answer

We need to follow the POE. 
A. f(0.25) = 0.5 , So f(f(0.25)) = is not a natural number.
B. 0.04 => Similarly f(f(0.04)) = is not a natural number
C. 0.016 => f(0.016) itself is not a natural number. 
D. 0.008=> Same as C. 
E. 0.0016 => f(0.0016)=0.04. 
f(f(0.0016)) = f(0.04) => 0.2 => 1/5. Reciprocal of 5(prime number) . 
 
Hence IMO E

hr1212 · Answer

f(x) = \sqrt{x}

f(f(x)) = \sqrt{f(x)} = \sqrt {x}

Checking the options -

0.0016 =  2^{4}*10^{-4}  =  \sqrt { 2^{4}*10^{-4}}  =  2 * 10^{-1}  =  \frac{2}{10}  =  \frac{1}{5}

Reciprocal is a prime number - 5

Answer : E

Krunaal · Answer

f(x) = (x)^1/2; f(f(x)) = (x)^1/4

Let reciprocal of a prime no. be 1/p, where p is a prime no.

(x)^1/4 = 1/p

x = 1/(p)^4

Let's test the answer choices:

a. 0.25 = 1/4 = 1/(2)^2 This does not equate to 1/(p)^4  INCORRECT 
b. 0.04 = 1/25 = 1/(5)^2 This does not equate to 1/(p)^4  INCORRECT 
c. 0.016 = 2/125 This does not equate to 1/(p)^4  INCORRECT 
d. 0.008 = 1/125 = 1/(5)^3 This does not equate to 1/(p)^4  INCORRECT 
e. 0.0016 = 1/625 = 1/(5)^4 This equates to 1/(p)^4  CORRECT

Answer E.

nikiki · Answer

f(f(x)) = number^1/4

Lets see the options
(A) 0.25 = 1/4
1st square root = 1/2
2nd square root = 1/sqrt(2)
Not a prime numer

(B) 0.04
1st square root = 0.2 = 1/5
2nd square root = 1/sqrt(5)
not a prime number

(C) 0.016 = 2/125
1st square root = ...
2nd square root = ..
clearly not a prime number

(D) 0.008
1st square root = 
2nd square root = 
clearly not a prime number

(E) 0.0016
1st square root = 0.04
2nd square root = 0.2 = 1/5
 5 is the prime number

#when we take a glance at the options, only E will give a simple quad root

Navya442001 · Answer

Understand the function: f(x) = √x. Therefore, f(f(x)) = √(√x) = x^(1/4).

Set up the equation: We're told that f(f(x)) is the reciprocal of a prime number. Let p be a prime number. Then:

x^(1/4) = 1/p

Solve for x: Raise both sides to the power of 4:

x = (1/p)^4 = 1/p^4

Test the answer choices: We need to find which of the given values of x can be expressed as 1/p4, where p is a prime number.

A. 0.25 = 1/4 = 1/(2^2) ≠ 1/2^4. A is not a possible solution.
B. 0.04 = 1/25 = 1/(5^2) ≠ 1/p^4. B is not a solution.
C. 0.016 = 16/1000 = 2/125 = 2/(5^3) ≠ 1/p4. C is not a solution.
D. 0.008 = 8/1000 = 1/125 = 1/(5^3) ≠ 1/p4. D is not a solution.
E. 0.0016 = 16/10000 = 1/625 = 1/(5^4). Here, p=5, which is prime. So, E is a possible solution.
Therefore E could be the value of x.

Lizaza · Answer

In this case, we're looking for the root of fourth power of X, which must have a prime number in the denominator.

From the fractions provided, only   0.0016=\frac{1}{625}=\frac{1}{5^4}  gives us the result, as the denominator will be 5 (prime).
  The answer is therefore E.

12 Days of Christmas GMAT Competition - Day 6: If x is a positive

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

12 Days of Christmas GMAT Competition - Day 6: If x is a positive

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards