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Is the positive integer z a prime number? (1) z and the square root

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Bunuel
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Is the positive integer z a prime number? (1) z and the square root [#permalink] New post   27 Oct 2021, 04:00
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49% (01:49) correct 51% (01:54) wrong based on 107 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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B
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Re: Is the positive integer z a prime number? (1) z and the square root [#permalink] New post   27 Oct 2021, 07:46
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.
Let us take z =2 and y= 9
Factors of z = 1,2
Factors of √9 = 1,3
Here z is prime no.

Let us take another example where z= 9 and y =16
Factors of z = 1,3,9
Factors of √16 = 1,2,4
Here z is not prime no.

Not sufficient.

(2) z and the perfect square y have the same number of unique factors.
y being a perfect square has 3 factors.
like 25 has factors as 1,5,25
But a prime no cannot have more than 2 factors
Sufficient

IMO B
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Re: Is the positive integer z a prime number? (1) z and the square root [#permalink] New post   18 Aug 2022, 04:27
I have an inquiry regarding this question. Statement (2), what if y=√2， then z could be 2. Since statement (2) does not state that "integer y". Instead, it only said that "the perfect square y"
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Re: Is the positive integer z a prime number? (1) z and the square root [#permalink] New post   27 Aug 2022, 09:45
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ThatDudeKnows.

A little guidance here please.

Isn't 1 a perfect square as well?
In this case, number of unique factors will be only 1. Then how can B give a definite solution?
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Re: Is the positive integer z a prime number? (1) z and the square root [#permalink] New post   29 Aug 2022, 02:06
Expert Reply
Bunuel 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.



Project DS Butler Data Sufficiency (DS3)


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Solution:

Pre Analysis:

We are asked if positive integer \(z\) is prime or not. We know prime numbers are positive integers with only 2 factors.


Statement 1: z and the square root of integer y have the same number of unique prime factors

Let's take 2 cases:

Case 1: \(z=2\) and \(\sqrt{y}=\sqrt{25}=5\) both have one unique prime factor i.e., 2 and 5 itself. In this case, \(z=2\) is prime
Case 2: \(z=10\) and \(\sqrt{y}=\sqrt{10}=10\) both have two unique prime factors i.e., 2 and 5. In this case, \(z=10\) is not a prime

Thus, statement 1 alone is not sufficient and we can eliminate options A and D


Statement 2: z and the perfect square y have the same number of unique factors

There are 2 cases here:

Case 1: \(y=1\) (perfect square) has one factor (1 itself). If z also has one factor, then we can be sure that z is not a prime number
Case 2: \(y>1\) (perfect square) i.e., every other perfect square like 4, 9, 16, etc have least three factors. If z has 3 or more factors, then we can be sure that z is not a prime number

Thus, statement 2 alone is sufficient to answer that z is not a prime number


Hence the right answer is Option B
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Re: Is the positive integer z a prime number? (1) z and the square root [#permalink]
29 Aug 2022, 02:06
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Bunuel
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