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

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If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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13 Dec 2012, 09:35
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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
Math Expert
Joined: 02 Sep 2009
Posts: 55271
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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13 Dec 2012, 09:39
32
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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

Since $$3 < x < 100$$, then $$1<\frac{x}{3}<33\frac{1}{3}$$ (just divide all parts of the inequality by 3).

$$\frac{x}{3}$$ should be the square of a prime number, thus $$\frac{x}{3}$$ could be 2^2=4, 3^2=9, or 5^2=25.

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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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19 Dec 2012, 07:19
12
3
For x/3 to be a square of a prime number, x = (prime no)^2 x 3

List squares of some prime nos:
4,9,25,49

x can be:

4x3, 9x3 and 25x3

Anything above 25x3 will make x>100, but we are given that 3<x<100

Hence, 3 nos, Option B.
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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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13 Dec 2012, 09:48
3
1
x/3 is a square or we can take x/3 as y^2; x = y^2 * 3.
Since x should be between 3 and 100, y^2 can take squares 4, 9 and 25.
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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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19 Dec 2012, 07:26
1
1
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

I think Bunnuel has given a perfect explanation that needs no further detailing.

I just want to highlight a point in these types of questions. For the questions that ask "how many values of x".. they will usually be of consecutive numbers. But when the numbers asked are primes, then it is manual counting that is required because prime numbers do not follow any standard pattern. They will definitely be countable and the total will usually be less than 25 in GMAT.
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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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08 Jul 2013, 03:58
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

I got confused in this question...why are we considering values 2 and 3 when it is given that 3<x<100...
Math Expert
Joined: 02 Sep 2009
Posts: 55271
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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08 Jul 2013, 04:31
4
1
anu1984 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

I got confused in this question...why are we considering values 2 and 3 when it is given that 3<x<100...

The question is: for how many values of x is x/3 the square of a prime number.

The answer is: for three values of x (namely for 12, 27, and 75) x/3 is the square of a prime number.

Hope it's clear.
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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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08 Jul 2013, 05:12
Bunuel wrote:
anu1984 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

I got confused in this question...why are we considering values 2 and 3 when it is given that 3<x<100...

The question is: for how many values of x is x/3 the square of a prime number.

The answer is: for three values of x (namely for 12, 27, and 75) x/3 is the square of a prime number.

Hope it's clear.

Yes got it now...thanks..
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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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05 Apr 2014, 15:40
Bunuel 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

Since $$3 < x < 100$$, then $$1<\frac{x}{3}<33\frac{1}{3}$$ (just divide all parts of the inequality by 3).

$$\frac{x}{3}$$ should be the square of a prime number, thus $$\frac{x}{3}$$ could be 2^2=4, 3^2=9, or 5^2=25.

Going through the explanation, I get *why* the answer is 3 but unfortunately, I wouldn't have been able to come to that solution on my own.

If I treat this problem as substitution, i can sub x=Prime^2 * 3 and then solve but why did you take the other route? Why did you divide both sides of the inequality by 3?
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Posts: 55271
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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05 Apr 2014, 16:18
russ9 wrote:
Bunuel 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

Since $$3 < x < 100$$, then $$1<\frac{x}{3}<33\frac{1}{3}$$ (just divide all parts of the inequality by 3).

$$\frac{x}{3}$$ should be the square of a prime number, thus $$\frac{x}{3}$$ could be 2^2=4, 3^2=9, or 5^2=25.

Going through the explanation, I get *why* the answer is 3 but unfortunately, I wouldn't have been able to come to that solution on my own.

If I treat this problem as substitution, i can sub x=Prime^2 * 3 and then solve but why did you take the other route? Why did you divide both sides of the inequality by 3?

You should use whichever approach suits you the best and gives the correct answer in minimum time.

As for my solution, I divided by 3 because this way I directly get the range for x/3 ($$1<\frac{x}{3}<33\frac{1}{3}$$), and it becomes easier to evaluate the number of values for it.
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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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07 Apr 2014, 01:18
12
2
Setting up the equation as follows:

$$\frac{x}{3}$$ should be a square

So, say $$\frac{x}{3} = a^2$$

$$x = 3a^2$$

$$3 < 3 a^2 < 100$$

For a = 1; the condition fails

For a = 2; x = 12

For a = 3; x = 27

For a = 5; x = 75

For a = 7; condition fails

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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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09 May 2014, 15:40
Bunuel wrote:
russ9 wrote:
Bunuel wrote:
Going through the explanation, I get *why* the answer is 3 but unfortunately, I wouldn't have been able to come to that solution on my own.

If I treat this problem as substitution, i can sub x=Prime^2 * 3 and then solve but why did you take the other route? Why did you divide both sides of the inequality by 3?

You should use whichever approach suits you the best and gives the correct answer in minimum time.

As for my solution, I divided by 3 because this way I directly get the range for x/3 ($$1<\frac{x}{3}<33\frac{1}{3}$$), and it becomes easier to evaluate the number of values for it.

Hi Bunuel,

Can you perhaps recommend a few similar problems?

Thanks!
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Posts: 55271
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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10 May 2014, 05:18
russ9 wrote:
Bunuel wrote:
russ9 wrote:

You should use whichever approach suits you the best and gives the correct answer in minimum time.

As for my solution, I divided by 3 because this way I directly get the range for x/3 ($$1<\frac{x}{3}<33\frac{1}{3}$$), and it becomes easier to evaluate the number of values for it.

Hi Bunuel,

Can you perhaps recommend a few similar problems?

Thanks!

Check this one: for-how-many-values-of-k-is-12-12-the-least-common-multiple-86737.html
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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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12 Sep 2014, 06:12
1
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

x/3 the square of a prime number ; 3 < x < 100..

divide the above by 3 then 1 < x/3 < 33..

so x/3 could be 4, 9, 16 25 .. since 16 is a square of 4 , which is not a prime answer should be 3.
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If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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30 Oct 2014, 01:13
1
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

What the question stem asks us is how many numbers within 100 are 3 times the square of a prime number?

(2^2)* 3 = 12 - less than 100 and greater than 3
(3^3)* 3 = 27 - less than 100
(5^5)* 3 = 75 - less than 100

We don't have to consider next prime 7 since (7*7)*3 > 100

Thus only 3 possible numbers.
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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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15 Jun 2015, 18:29
I got B

x/3 = (prime)^2
x = 3(prime) ^2

The only values for x that are 3 < x < 100 are if the prime numbers are 2,3 and 5. Once you hit 7 your x value becomes greater than 100. Therefore there are only 3 values for x.
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Re: If 3 < x < 100, for how many values of x is the square of a prime numb  [#permalink]

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12 Sep 2015, 00:31
X < 100

So, square root of x has to be less than 10.

Also, square is positive, square root has to be positive (as prime numbers are never negative).

The following numbers between 1 and 10 are prime: 2, 3, 5, 7.

Their squares will lie between 4 and 100.

So, (C).
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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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13 Sep 2015, 21:12
nahid78 wrote:
If 3 < x < 100, for how many values of x is the square of a prime number?

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

Hi nahid78,

The values would be 4, 9, 25, 49, 81

But if you have mistyped the question and we are required to find the values of (x/3)
then the values would be 12, 27, 75. Three values
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If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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Updated on: 08 Feb 2018, 16:07
1
I can see two approaches for this:

Mathematical:

$$\frac{x}{3}$$ needs to be a square prime

So,
$$\frac{x}{3}=p^2$$

$$x=3p^2$$

$$3<3p^2<100$$

$$1<p^2<33.33$$
Therefore:
$$1<p<6$$
The only primes between 1 and 6 are: 2,3,5

Logical:

$$x$$ needs to be completely divided by 3, resulting in a square of a prime.

$$\frac{3*7^2}{3}$$---- Prime can't be seven, it would make x>100... so answer is 2,3,5
$$\frac{3*5^2}{3}$$
$$\frac{3*3^2}{3}$$
$$\frac{3*2^2}{3}$$

+Kudos if you think this helped. Thanks!

Originally posted by BlacknYellow on 28 Jan 2016, 14:44.
Last edited by BlacknYellow on 08 Feb 2018, 16:07, edited 1 time in total.
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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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25 May 2016, 22:18
3
Attached is a visual that should help.
Attachments

Screen Shot 2016-05-25 at 9.43.06 PM.png [ 104.41 KiB | Viewed 19530 times ]

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Re: If 3 < x < 100, for how many values of x is x/3 the square   [#permalink] 25 May 2016, 22:18

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