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Manager  Joined: 12 Feb 2012
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If x^(1/3) is a positive integer, does x^(1/3) have more than two dist  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 60% (01:40) correct 40% (01:45) wrong based on 77 sessions

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If $$\sqrt{x}$$ is a positive integer, does $$\sqrt{x}$$ have more than two distinct integer factors?

(1) 64 < x < 216

(2) x is divisible by 5.

1) 64^(1/3)=4 , 125^(1/3)=5, and 216^(1/3)=6. Since x^(1/3) is an integer, x must equal 5. 5 is a prime number and only has two factors 1 and 5. So the answer is NO. Hence sufficient.

2) X=5k. Well 1000=5*200. x^(1/3)=1000^(1/3)=10 , which has 1,2,5,10 as its factors. So the answer the question is YES.
But we can also pick x=125=5*25. x^(1/3)=125^(1/3)=5, which has only two factors: 1 and 5. Hence the answer is NO. Hence INSUFF.
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Re: If x^(1/3) is a positive integer, does x^(1/3) have more than two dist  [#permalink]

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b. x can be 125 or 125^3
thus x ^ (1/3) = 5 or 125, factors 1,5 or 1,5,25,125 so on.

hence Insufficient.

A it is.
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Re: If x^(1/3) is a positive integer, does x^(1/3) have more than two dist  [#permalink]

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If $$\sqrt{x}$$ is a positive integer, does $$\sqrt{x}$$ have more than two distinct integer factors?

(1) 64 < x < 216 --> take the cube root: $$4<\sqrt{x}<6$$ --> since given that $$\sqrt{x}=integer$$ then: $$\sqrt{x}=5$$. 5 has only 2 distinct positive factors: 1 and 5. Sufficient.

(2) x is divisible by 5 --> if $$\sqrt{x}=5$$ the answer is NO but if $$\sqrt{x}=10$$ the answer is YES. Not sufficient.

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Re: If x^(1/3) is a positive integer, does x^(1/3) have more than two dist  [#permalink]

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alphabeta1234 wrote:
If x^(1/3) is a positive integer, does x^(1/3) have more than two distinct integer factors?

(1) 64 < x < 216
(2) x is divisible by 5.

Cannot Find an OA: My Solution

1) 64^(1/3)=4 , 125^(1/3)=5, and 216^(1/3)=6. Since x^(1/3) is an integer, x must equal 5. 5 is a prime number and only has two factors 1 and 5. So the answer is NO. Hence sufficient.

2) X=5k. Well 1000=5*200. x^(1/3)=1000^(1/3)=10 , which has 1,2,5,10 as its factors. So the answer the question is YES.
But we can also pick x=125=5*25. x^(1/3)=125^(1/3)=5, which has only two factors: 1 and 5. Hence the answer is NO. Hence INSUFF.

Another way to look at the question.
Basically question says Is x^(1/3) a prime number, if x^(1/3) is a positive integer?
(1) x=5, which is a prime. -----> Sufficient
(2) x=5k, where is a positive integer -----> x= 5,10....----->Insufficient

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Re: If x^(1/3) is a positive integer, does x^(1/3) have more than two dist  [#permalink]

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A

1) x=125 .so x^1/3 =5 only 1&5 as factors.Suff

2)x^1/3 can be 25 or 5 which has more than 2 and only 2 factors.Not Suff
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Re: If x^(1/3) is a positive integer, does x^(1/3) have more than two dist  [#permalink]

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alphabeta1234 wrote:
If x^(1/3) is a positive integer, does x^(1/3) have more than two distinct integer factors?

(1) 64 < x < 216
(2) x is divisible by 5.

x^1/3 is prime or not?

(1) 64 < x < 216
64^1/3 < x^1/3 < 216 ^1/3
4<< x^1/3 <6

Since x^1/3 is an integer it will be 5 (prime). sufficient

(2) x is divisible by 5. Well it can be 5 or any multiple of 5. not sufficient.

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Re: If x^(1/3) is a positive integer, does x^(1/3) have more than two dist  [#permalink]

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_________________ Re: If x^(1/3) is a positive integer, does x^(1/3) have more than two dist   [#permalink] 05 Apr 2019, 00:27
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