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

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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
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

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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# Scott Woodbury-Stewart

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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
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
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?
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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
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.

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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
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

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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# Jeffrey Miller

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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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31 Oct 2017, 15:33
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, 12:09
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
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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, 12:28
1
dave13 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

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

Hi dave13

Square of a prime number means that the resulting square has prime number as its root.

For e.g. 9 has 3 as its root so 9 is square of a prime number because 3 is prime
But 16 is not a square of a prime number because it’s root is 4 which is not prime

Posted from my mobile device
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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 Sep 2018, 06:38
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:
$$2^2=4$$, $$3^2=9$$, $$5^2=25$$

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

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28 Jan 2019, 19:58
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:

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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, 09:09
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.

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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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11 May 2019, 05:20
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, 05:20

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