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Manager  Joined: 02 Dec 2012
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If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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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 V
Joined: 02 Sep 2009
Posts: 58445
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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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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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

Answer = 3 = 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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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.
Intern  Joined: 26 May 2012
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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
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.
Manager  Joined: 08 Apr 2012
Posts: 114
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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2
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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Intern  Joined: 28 Feb 2012
Posts: 2
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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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

I got confused in this question...why are we considering values 2 and 3 when it is given that 3<x<100...
Math Expert V
Joined: 02 Sep 2009
Posts: 58445
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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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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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..
Manager  Joined: 15 Aug 2013
Posts: 228
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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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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Joined: 02 Sep 2009
Posts: 58445
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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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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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!
Math Expert V
Joined: 02 Sep 2009
Posts: 58445
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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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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Manager  Joined: 07 Apr 2014
Posts: 100
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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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

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.
Intern  Joined: 18 Oct 2014
Posts: 1
If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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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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Posts: 4
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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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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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.

Hence, answer is 4.

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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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,

If your question asks how many values of x, then the answer will be 5
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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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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Attached is a visual that should help.
Attachments Screen Shot 2016-05-25 at 9.43.06 PM.png [ 104.41 KiB | Viewed 25638 times ]

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