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x is a positive integer less than

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x is a positive integer less than  [#permalink]

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New post Updated on: 13 Aug 2018, 02:25
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Difficulty:

  55% (hard)

Question Stats:

64% (02:34) correct 36% (02:02) wrong based on 136 sessions

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Question:



\(x\) is a positive integer less than \(30\), which of the following option correctly indicates the number of values of \(x\) for which \(x^2/18\) has \(2\) factors?
A) 0
B) 1
C) 4
D) 5
E) 7

This is

Question 9 of The e-GMAT Number Properties Marathon




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Question 10 of the Marathon


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Originally posted by EgmatQuantExpert on 28 Feb 2018, 02:56.
Last edited by EgmatQuantExpert on 13 Aug 2018, 02:25, edited 3 times in total.
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Re: x is a positive integer less than  [#permalink]

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New post 28 Feb 2018, 03:28
EgmatQuantExpert wrote:

Question:



\(x\) is a positive integer less than \(30\), which of the following option correctly indicates the number of values of \(x\) for which \(x^2/18\) has \(2\) factors?
A) 0
B) 1
C) 4
D) 5
E) 7


1=< x < 30

for \(x^2/18\) to have 2 factors x^2 greater than 18

5<= x < 30

now x can't be odd as 18 = even x odd and x can't be only even i.e 16,8 etc

so from 5 to 30 remove odd values and purely even value( powers of 2 )

6 10 12 14 18 20 22 24 26 28

here numbers must be multiple of 3 as 18 = 9x2 and if x has 3 x^2 will have 9

so

6 12 18 24 left

6^2 = 36 and 36/18 = 2 ( factors 2)
12^2 = 144 and 144/18 = 8 factors 4
18^2/18 = 18 factors 6
24^2/18 = 4/3x24 = 32 2^5 factors 6

x can only be 6
(B) imo
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Re: x is a positive integer less than  [#permalink]

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New post 28 Feb 2018, 12:23
1
2

Solution:



    \(x^2/(2×3^2 )\) = Prime number
    \(x^2= 2×3^2×\)prime number………………(1)
Now, Right-hand side of the equation (1) is a perfect square. Hence, the only possible prime number that will make left-hand side of the equation a perfect square is 2 only.
Therefore,
    \(x^2= 2*3^2 *2\)
    \(x= 2*3\)
    \(x= 6\)
Hence, only \(1\) value of x is possible.
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Re: x is a positive integer less than  [#permalink]

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New post 06 Apr 2019, 21:38
Given x2/18 has 2 factors

That means the x2 when divided by 18 will result in a prime number. Because a prime number has 2 factors.

18=2*3^2

x2=2*3^2*k where value of k should be such that it RHS becomes a perfect square.


Put k=2 . Therefore X2=2^2*3^2 and hence x=6.

Now if you put k=(2^3*3^2) , x2= 2^4*3^4 and therefore x will be 36 which is not possible since x<30

Therefore we have only one value of x

IMO B.
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Re: x is a positive integer less than   [#permalink] 06 Apr 2019, 21:38
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