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

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Manager
Joined: 15 Aug 2013
Posts: 223
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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09 May 2014, 14:40
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!
Math Expert
Joined: 02 Sep 2009
Posts: 62368
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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10 May 2014, 04:18
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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Posts: 96
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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12 Sep 2014, 05:12
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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Joined: 16 Feb 2015
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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, 17:29
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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11 Sep 2015, 23:31
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, 20:12
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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Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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13 Sep 2016, 22:33
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

Assume x=+int
mistake mine: (x/3)^2=P wrong
(x/3) = P^2
Director
Joined: 04 Sep 2015
Posts: 673
Location: India
WE: Information Technology (Computer Software)
If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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25 Mar 2017, 23:28
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

The question confusingly adds x/3... lol so the square is the value of x/3

so we take first prime numbers 2,3,5,7,11 and are 4,9,25,49,121..

Since these values should be x/3 and the value of x will be 4*3,9*3,25*3,49*3..... which are 12,27,75,147....

we only need values below 100 so we have only 12,27 and 75 Three is the answer.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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15 Apr 2017, 13:20
1
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.

How do you know to divide the inequality by 3. is there a general strategy for this?
VP
Joined: 07 Dec 2014
Posts: 1240
If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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15 Apr 2017, 16:37
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

let x/3=y^2
3y^2<100
y^2<34
y^2 can only=4, 9, or 25
3
B
Intern
Joined: 14 May 2016
Posts: 17
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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31 Oct 2017, 14:33
I did an excel spreadsheet to understand this. However, the explanation from other people to divide 100/3 = 33.33... helps to reduce the amount of work.

thank you all!
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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 2018, 11:09
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

Hello niks18

one question: "square of a prime number" is it the same as "square number" ? just dont get the meaning of this wording "square of a prime number"

for example 9 is square number because 3*3 =9

thanks
Intern
Joined: 15 Sep 2018
Posts: 31
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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25 Sep 2018, 05:38
First, we need to find the range of values for $$\frac{x}{3}$$.

Given that $$3<x<100,$$ we can divide all sides of the inequality and have

$$\frac{3}{3}<\frac{x}{3}<\frac{100}{3}$$
$$1<\frac{x}{3}<33\frac{1}{3}$$

With this range, there are only a few numbers that are squares of a prime number. They are:
$$2^2=4$$, $$3^2=9$$, $$5^2=25$$

Manager
Joined: 19 Jan 2018
Posts: 89
If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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28 Jan 2019, 18:58
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

Translation:
How many values of x is x/3 the square of a prime number

$$\frac{X}{3}$$=$$p^2$$, p being a prime number

Rearrange the formula so that it equals to x:

$$X = p^2 *3$$

Now test prime numbers out:

$$X = 1^2 *3$$ X = 3 (Discard p=1 because X has to be larger than 3, not equal to it)

$$X = 2^2 *3$$ X = 12

$$X = 3^2 *3$$ X = 27

$$X = 5^2 *3$$ X = 75

$$X = 7^2 *3$$ X = 149 (Discard p=7 because x>100)

In total 3 Primes fit the condition:

Senior Manager
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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 Apr 2019, 08:09
Have to read more carefully... wasted almost a minute thinking the question said the the prime number must be >3 rather than x (which would mean only 5 fits).

An easy way to do this is to revert the equation... x = prime#² * 3
Since x < 100, we can just test
2²*3 = 12
3²*3 = 27
5²*3 = 75
7²* 3 = 147, too big.

Manager
Joined: 25 Sep 2018
Posts: 65
Re: If 3 < x < 100, for how many values of x is x/3 the square  [#permalink]

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11 May 2019, 04:20
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

Here,
x/3=p^2
=>x=3p^2

3*2^2=12 possible
3*3^2=27 possible
3*5^2=75 possible
But,3*7^2=147 not possible

So, x has 3 possible values

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Re: If 3 < x < 100, for how many values of x is x/3 the square   [#permalink] 11 May 2019, 04:20

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