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

Question

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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asishron29181 · Answer

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

cjhyou700 · Answer

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"

poorvi125 · Answer

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?

SaquibHGMATWhiz · Answer

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

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GMAT Data Sufficiency (DS) Questions

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

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