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If x^(1/3) is a positive integer, does x^(1/3) have more than two dist

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alphabeta1234
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If x^(1/3) is a positive integer, does x^(1/3) have more than two dist [#permalink] New post   08 Apr 2012, 19:36
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If \(\sqrt[3]{x}\) is a positive integer, does \(\sqrt[3]{x}\) have more than two distinct integer factors?


(1) 64 < x < 216

(2) x is divisible by 5.

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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] New post   08 Apr 2012, 22:51
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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] New post   09 Apr 2012, 00:58
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If \(\sqrt[3]{x}\) is a positive integer, does \(\sqrt[3]{x}\) have more than two distinct integer factors?

(1) 64 < x < 216 --> take the cube root: \(4<\sqrt[3]{x}<6\) --> since given that \(\sqrt[3]{x}=integer\) then: \(\sqrt[3]{x}=5\). 5 has only 2 distinct positive factors: 1 and 5. Sufficient.

(2) x is divisible by 5 --> if \(\sqrt[3]{x}=5\) the answer is NO but if \(\sqrt[3]{x}=10\) the answer is YES. Not sufficient.

Answer: A.
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Re: If x^(1/3) is a positive integer, does x^(1/3) have more than two dist [#permalink] New post   13 Sep 2012, 04:04
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

Show Spoiler
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

Answer A
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Re: If x^(1/3) is a positive integer, does x^(1/3) have more than two dist [#permalink] New post   12 Jun 2016, 08:10
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] New post   13 Jun 2016, 09:26
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.

A is the answer
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Re: If x^(1/3) is a positive integer, does x^(1/3) have more than two dist [#permalink] New post   05 Jul 2023, 23:49
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Re: If x^(1/3) is a positive integer, does x^(1/3) have more than two dist [#permalink]
05 Jul 2023, 23:49
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