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# Is the positive integer z a prime number?

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Director
Joined: 11 Feb 2015
Posts: 711
Is the positive integer z a prime number?  [#permalink]

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28 Jan 2019, 07:52
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1
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Difficulty:

55% (hard)

Question Stats:

61% (01:38) correct 39% (01:25) wrong based on 33 sessions

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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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Director
Joined: 09 Mar 2018
Posts: 912
Location: India
Is the positive integer z a prime number?  [#permalink]

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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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Intern
Joined: 24 Jan 2013
Posts: 14
Re: Is the positive integer z a prime number?  [#permalink]

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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

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