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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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New post 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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New post 12 Sep 2014, 06:12
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Walkabout 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



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

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New post 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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New post 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.

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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New post 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,

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

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New post 13 Sep 2016, 23:33
Walkabout 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

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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New post 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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New post 15 Apr 2017, 14:20
1
Bunuel wrote:
Walkabout 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.

Answer: B.


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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New post 15 Apr 2017, 17:37
Walkabout 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


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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New post 30 Aug 2017, 14:45
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Walkabout 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


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.
Answer:

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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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New post 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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New post 09 May 2018, 12:09
Walkabout 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 :-)
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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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New post 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:
\(2^2=4\), \(3^2=9\), \(5^2=25\)

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

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New post 28 Jan 2019, 19:58
Walkabout 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



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:

Answer is 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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New post 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.

So the answer is 3.
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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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New post 11 May 2019, 05:20
Walkabout 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

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

Answer is B

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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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