If 3 < x < 100, for how many values of x is x/3 the sq

rsarashi wrote:If 3 < x < 100, for how many values of x is x/3 the square of a prime number?

(A) Two
(B) Three
(C) Four
(D) Five
(E) Nine

OAB

We have a situation: x/3 the square of a prime number such that 3 < x < 100.

3 < x < 100 => 1 < x/3 < 100/3 => 1 < x/3 < 33.33 => 1 < p^2 < 33.33; where p is a prime number

Let us list down few prime numbers; they are 2, 3, 5, 7, 11, etc.

Since x/3 is a square of a prime number, only prime numbers that would qualify are 2, 3, & 5.

The correct answer: B

Hope this helps!

Relevant book: Manhattan Review GMAT Number Properties Guide

-Jay
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rsarashi wrote:If 3 < x < 100, for how many values of x is x/3 the square of a prime number?

(A) Two
(B) Three
(C) Four
(D) Five
(E) Nine

OAB

We want values of x (where 3 < x < 100) such that x/3 is the square of a prime number.
So, let's start checking squares of prime numbers.
Some prime numbers are 2, 3, 5, 7, 11, etc

2² = 4 and (3)(4) = 12. So, x = 12 meets the given conditions.
3² = 9 and (3)(9) = 27. So, x = 27 meets the given condition12
5² = 25 and (3)(25) = 75. So, x = 75 meets the given conditions.
7² = 49 and (3)(49) = 147. No good. We need values of x such that 3 < x < 100

So, there are exactly 3 values of x that meet the given conditions.
Answer: B

Cheers,
Brent

rsarashi wrote:

2² = 4 and (3)(4) = 12. So, x = 12 meets the given conditions.
3² = 9 and (3)(9) = 27. So, x = 27 meets the given condition12
5² = 25 and (3)(25) = 75. So, x = 75 meets the given conditions.
7² = 49 and (3)(49) = 147. No good. We need values of x such that 3 < x < 100

Hi Brent ,

Thank you so much for your reply.

Just a quick question can you please advise that why did you multiply with 3? We have to divide x/3.

Please explain sir.

Great question.

First off, we want x/3 to be the square of a prime number.
Since all prime numbers are integers, the square of a prime number will be an integer.

In order for x/3 to be an integer, it must be the case that x is divisible by 3.
Another way to put it is that x must be a multiple of 3.
This why I took each squared value and multiplied it by 3.

Cheers,
Brent

Hi All,

We’re told that 3 < X < 100. We’re asked for the number of possible values for X that would make X/3 the SQUARE of a PRIME NUMBER. From the answer choices, we know that there are at least 2 values, but no more than 9 values, that fit what we’re asked for, so we should be able to list them all out without too much difficulty.

To start, it’s worth noting that since X < 100, we know that the value of X/3 will be 33 or less. We’re looking for SQUARES of PRIME NUMBERS, so the number of possibilities is going to be really ‘limited’; we can start at the smallest Prime and work up…

2; 2^2 = 4…. If X=12, then 12/3 = 4 which is a square of a prime

3: 3^2 = 9… If X = 27, then 27/3 = 9 which is a square of a prime

5: 5^2 = 25… If X = 75, then 75/3 = 25 which is a square of a prime

7: 7^2 = 49… Here, we would need X to be GREATER than 100 (specifically 147, which is not allowed).

Thus, there are only 3 possible values of X that fit the ‘restrictions’ in the prompt.

Final Answer: B

GMAT Assassins aren’t born, they’re made,
Rich