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Is the positive integer z a prime number?

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Director
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Is the positive integer z a prime number?  [#permalink]

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New post 28 Jan 2019, 07:52
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Question Stats:

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Is the positive integer z a prime number?

(1) z and the square root of integer y have the same number of unique prime factors.

(2) z and the perfect square y have the same number of unique factors.

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Is the positive integer z a prime number?  [#permalink]

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New post 28 Jan 2019, 08:24
CAMANISHPARMAR wrote:
Is the positive integer z a prime number?

(1) z and the square root of integer y have the same number of unique prime factors.

(2) z and the perfect square y have the same number of unique factors.


1) z and the square root of integer y have the same number of unique prime factors.

z= 3 and \(\sqrt{9}\), will satisfy this. 3 and 3(root 9), have 1 unique factor
Will answer the question as Yes

z =14(2*7) and \(\sqrt{196}\), will satisfy the statement. 2,7 and root 196(will give 2*7) will have 2 unique factors
Will answer the question as No

(2) z and the perfect square y have the same number of unique factors.
perfect square y can be 1,4,9,25,36,49,64

doesn't this mean they should be equal ??, only they can have same number of unique factorss

Examples 4 and 4, 8 and 16 No this wont have

z is a +ive Integer

But then z can be 1
Will answer the question as No

z can be 4
Will answer the question as No

A definite No.

IMO B
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Re: Is the positive integer z a prime number?  [#permalink]

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New post 28 Jan 2019, 09:28
1
1.
say y = 9, sqrt of y = 3 , NUMBER OF UNIQUE PRIME FACTORS =1

Now z to also have 1 number of unique prime factors , Z can be either -
3 ( PRIME )
or
9 ( NON PRIME)

Not sufficient

2.
y is a perfect square say a^2 form
Total Number of factors = 3 ( always an odd number)
For z to have same odd number of factors it can never be a prime number , because prime numbers always have only 2 factors ( 1 and number itself)
sufficient

HENCE B is answer
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Re: Is the positive integer z a prime number?   [#permalink] 28 Jan 2019, 09:28
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