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# x is a positive integer less than

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e-GMAT Representative
Joined: 04 Jan 2015
Posts: 2324
x is a positive integer less than  [#permalink]

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Updated on: 13 Aug 2018, 01:25
00:00

Difficulty:

75% (hard)

Question Stats:

55% (02:35) correct 45% (02:00) wrong based on 111 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

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Originally posted by EgmatQuantExpert on 28 Feb 2018, 01:56.
Last edited by EgmatQuantExpert on 13 Aug 2018, 01:25, edited 3 times in total.
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Joined: 07 Jan 2016
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Re: x is a positive integer less than  [#permalink]

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28 Feb 2018, 02:28
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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Joined: 04 Jan 2015
Posts: 2324
Re: x is a positive integer less than  [#permalink]

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28 Feb 2018, 11:23
1

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.
_________________

Number Properties | Algebra |Quant Workshop

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Guillermo's Success Story | Carrie's Success Story

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Number Properties – Even Odd | LCM GCD | Statistics-1 | Statistics-2 | Remainders-1 | Remainders-2
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line | Inequalities

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

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Re: x is a positive integer less than &nbs [#permalink] 28 Feb 2018, 11:23
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# x is a positive integer less than

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