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
If 3 < x < 100, for how many values of x is x/3 the sq
This topic has expert replies
GMAT/MBA Expert
 Jay@ManhattanReview
 GMAT Instructor
 Posts: 3008
 Joined: Mon Aug 22, 2016 6:19 am
 Location: Grand Central / New York
 Thanked: 470 times
 Followed by:34 members
We have a situation: x/3 the square of a prime number such that 3 < x < 100.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
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
_________________
Manhattan Review GMAT Prep
Locations: New York  Hyderabad  Mexico City  Toronto  and many more...
Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.
GMAT/MBA Expert
 Brent@GMATPrepNow
 GMAT Instructor
 Posts: 16207
 Joined: Mon Dec 08, 2008 6:26 pm
 Location: Vancouver, BC
 Thanked: 5254 times
 Followed by:1268 members
 GMAT Score:770
We want values of x (where 3 < x < 100) such that x/3 is the square of a prime number.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
So, let's start checking squares of prime numbers.
Some prime numbers are 2, 3, 5, 7, 11, etc
2Â² = 4 and (3)(4) = 12. So, x = 12 meets the given conditions.
3Â² = 9 and (3)(9) = 27. So, x = 27 meets the given condition12
5Â² = 25 and (3)(25) = 75. So, x = 75 meets the given conditions.
7Â² = 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

 Master  Next Rank: 500 Posts
 Posts: 186
 Joined: Sat Dec 24, 2016 12:38 am
 Thanked: 5 times
 Followed by:3 members
Hi Brent ,2Â² = 4 and (3)(4) = 12. So, x = 12 meets the given conditions.
3Â² = 9 and (3)(9) = 27. So, x = 27 meets the given condition12
5Â² = 25 and (3)(25) = 75. So, x = 75 meets the given conditions.
7Â² = 49 and (3)(49) = 147. No good. We need values of x such that 3 < x < 100
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.
GMAT/MBA Expert
 Brent@GMATPrepNow
 GMAT Instructor
 Posts: 16207
 Joined: Mon Dec 08, 2008 6:26 pm
 Location: Vancouver, BC
 Thanked: 5254 times
 Followed by:1268 members
 GMAT Score:770
Great question.rsarashi wrote:Hi Brent ,2Â² = 4 and (3)(4) = 12. So, x = 12 meets the given conditions.
3Â² = 9 and (3)(9) = 27. So, x = 27 meets the given condition12
5Â² = 25 and (3)(25) = 75. So, x = 75 meets the given conditions.
7Â² = 49 and (3)(49) = 147. No good. We need values of x such that 3 < x < 100
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.
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

 Master  Next Rank: 500 Posts
 Posts: 186
 Joined: Sat Dec 24, 2016 12:38 am
 Thanked: 5 times
 Followed by:3 members
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.
Hi Brent ,
Perfect! All clear.
Thanks
GMAT/MBA Expert
 Jeff@TargetTestPrep
 GMAT Instructor
 Posts: 1462
 Joined: Thu Apr 09, 2015 9:34 am
 Location: New York, NY
 Thanked: 39 times
 Followed by:22 members
We can write out all the perfect squares below 100 that result from squaring a prime number. The prime numbers to consider are 2, 3, 5, and 7. The next prime number, 11, yields 121 when it is squared, which is too large, and so we only consider the following four squared prime numbers: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
4, 9, 25, 49
(Keep in mind that it's useful to have all the perfect squares below 100 memorized and note that 4 = 2^2, 9 = 3^2, 25 = 5^2, and 49 = 7^2.)
Next, we can write the question stem as an equation.
x/3 = (prime)^2
Solving for x, we have:
x = 3(prime)^2
From our list, we see that there are 3 values (4, 9, and 25) that, when we multiply them by 3, have a product that is less than 100: 3(4) = 12, 3(9) = 27, and 3(25) = 75. Thus, there are 3 values (12, 27, and 75) such that x/3 is the square of a prime number.
Answer: B
Jeffrey Miller
Head of GMAT Instruction
[email protected]
See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews
GMAT/MBA Expert
 [email protected]
 Elite Legendary Member
 Posts: 10392
 Joined: Sun Jun 23, 2013 6:38 pm
 Location: Palo Alto, CA
 Thanked: 2867 times
 Followed by:511 members
 GMAT Score:800
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
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