12 Days of Christmas GMAT Competition - Day 10: If 9

Question

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

If 9 < n < 99, for how many values of n is  \frac{1}{\sqrt{n}}  the reciprocal of a prime number?

A. Two
B. Three
C. Four
D. Five
E. Six

Bunuel · Accepted Answer

GMAT Club's Official Explanation:

We need such n from 9 < n < 99 that  \frac{1}{\sqrt{n}}  is the reciprocal of a prime number:

\frac{1}{\sqrt{n}} = \frac{1}{prime}

\sqrt{n} = prime

n = prime^2

From 9 to 99, not inclusive, there are two squares of primes: 5^2 = 25 and 7^2 = 49. Therefore, n can be 25 or 49.

Answer: A.

hr1212 · Answer

\frac{1}{\sqrt{n}} = \frac{1}{2} , \frac{1}{3} , \frac{1}{5} , \frac{1}{7}

Let's check against all the prime squares under 10

\sqrt{n}  = 2 or 3 or 5 or 7
n = 4 or 9 or 25 or49

Two of these values are less than 9, so after eliminating 4 and 9, we get only 2 possible values of n (25 and 49).

Answer: A

AbhiS101 · Answer

We are given that 9 < n < 99, and we want to find the number of values of  n  for which 1/√n is the reciprocal of a prime number. This means:
1/√n = 1/p, where  p  is a prime number.
Squaring both sides gives:
1/n = 1/p2
Therefore, n = p2.
Since 9 < n < 99, we have 9 < p2 < 99.
Taking the square root of all parts of the inequality gives:
√9 < p < √99
3 < p < 9.95 (approximately)
The prime numbers between 3 and 9.95 are 5 and 7.
If p = 5, then n = 52 = 25. If p = 7, then n = 72 = 49.
Both 25 and 49 are between 9 and 99. Therefore, there are two values of  n  that satisfy the given condition.

IMO A

Krunaal · Answer

The reciprocal of prime no.s are 1/2, 1/3, 1/5, 1/7, 1/11,............... and so on.

For 1/sqrt(n) to be a reciprocal of a prime no., sqrt(n) needs to be a prime no.

9 < n < 99 => 3 < sqrt(n) < sqrt(99)

Thus, sqrt(n) can take the prime no.s 5, and 7; and n = 25 and 49 => 2 values

Answer A.

Heix · Answer

https://gmatclub.com/forum/download/file.php?id=83388 GMAT-Club-Forum-bn91q510.jpeg

Navya442001 · Answer

For this 1/ √n to be the reciprocal of a prime number, √n must itself equal a prime number, which means n must equal the square of a prime number. Since 99 and (p^2)<99 The squares of primes are: 2^2=4 (too small, as 4≤9), 3^2=9 (excluded, as n>9), 5^2=25, 7^2=49, 11^2=121(too large, as n<99). The valid values of n are 25 and 49. For n=25: 1/ √ 25=1/5,and 5 is a prime number. For n=49: 1/ √ 49=1/7,and 7 is a prime number. Thus, there are two values of n: 25 and 49.

Sanjay9822 · Answer

n is between 9 and 99.
1/sqrt(n) is the reciprocal of a prime number.
Let p be the prime number.
1/sqrt(n) = 1/p
sqrt(n) = p
n = p^2
n must be a perfect square and between 9 and 99.
Prime numbers less than 10 are 2, 3, 5, 7.
Square these primes:
2^2 = 4 (not in range)
3^2 = 9 (in range)
5^2 = 25 (in range)
7^2 = 49 (in range)
Count valid values: 9, 25, 49.
Answer: 3

pintukr · Answer

If 9 < n < 99, for how many values of n is   1 /   √  n,    the reciprocal of a prime number?

A. Two 
 B. Three 
 C. Four 
 D. Five 
 E. Six

Given, 1 / √n is the reciprocal of a prime number;

√n is prime

For 9 < n < 99;....................................(1)

√n with n = 4; √n = 2 ....................................condition (1) not fulfilled  
 √n with n = 9; √n = 3 ....................................condition (1) not fulfilled 
  √n with n = 25; √n = 5 ....................................condition (1) fulfilled   
  √n with n = 49; √n = 7 ....................................condition (1) fulfilled   
 √n with n = 121; √n = 11 ....................................condition (1) not fulfilled

Only 2 possibilities, i.e. n =25, 49

(A) is the CORRECT answer

12 Days of Christmas GMAT Competition - Day 10: If 9 < n < 99, for how

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