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

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Intern
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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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28 Jan 2016, 14:44
1
KUDOS
I can see two approaches for this:

Mathematical:

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

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

$$x=3p^2$$

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

$$1<p^2<33$$ ($$P^2$$ can't be a non-int (100/3) )

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!

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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
1
KUDOS
Attached is a visual that should help.
Attachments

Screen Shot 2016-05-25 at 9.43.06 PM.png [ 104.41 KiB | Viewed 812 times ]

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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 2016, 06:04
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

An easy way to solve this problem is to first 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:

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 . Now solve for x.

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, the product will remain under 100: 3(4) = 12, 3(9) = 27 and 3(25) = 75. Thus, the answer is that there are 3 values (12, 27, and 75) such that x/3 is the square of a prime number.

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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, 23:33
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

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

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22 Nov 2016, 01:14
1
KUDOS
Great Official Question.
Here is what i did =>
x/2= P^2 where P is a prime number
Hence x=Prime^2*3

NOTE => Its a good idea to remember atleast the first Ten primes. Even Better if you know all the primes upto 100.
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
73
79
83
89
97
101

Now putting P=2,3,5 we get value of x in the bound (3,100)

Hence 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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25 Mar 2017, 12:09
since 3<x<100
then 1<x/3<33,3
then those numbers are 2, 3, 5
Hence 3 numbers

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

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26 Mar 2017, 00:28
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, 14:20
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?

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

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15 Apr 2017, 17:37
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

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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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30 Aug 2017, 14:45
Expert's post
Top Contributor
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

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 condition
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.
[Reveal] Spoiler:
B

RELATED VIDEO FROM OUR COURSE

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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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02 Sep 2017, 07:18
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

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:

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.

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Re: If 3 < x < 100, for how many values of x is x/3 the square   [#permalink] 02 Sep 2017, 07:18

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