Is the product of x and y a prime number?

Is the product of x and y a prime number?

(1) x^2 = 1
(2) y is positive and prime; x is positive but not prime

(C) 2008 GMAT Club - m01#30


My question is , why do we need to know whether the other non-prime number is 1 or not .
Why isn't statement 2 sufficient , that a prime * non-prime is always non-prime

Target question: Is the product xy a prime number?

Statement 1: x² = 1
This tells us that either x = 1 or x = -1, but it provides NO INFORMATION about y.
Consider these two possible cases:
Case a: x = 1 and y = 3. In this case, xy = (1)(3) = 3, which is prime. So, the answer to the target question is YES, the product xy IS prime
Case b: x = 1 and y = 6. In this case, xy = (1)(6) = 6, which is NOT prime. So, the answer to the target question is NO, the product xy is NOT prime
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y is positive and prime; x is positive but not prime
Let's test some possible cases again...
Case a: x = 1 (aside: 1 is NOT prime) and y = 3. In this case, xy = (1)(3) = 3, which is prime. So, the answer to the target question is YES, the product xy IS prime
Case b: x = 4 and y = 3. In this case, xy = (4)(3) = 12, which is NOT prime. So, the answer to the target question is NO, the product xy is NOT prime
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that either x = 1 or x = -1
Statement 2 tells us that x is POSITIVE, so we now know that x MUST equal 1
If x = 1, then the product xy = (1)(y) = y
Statement 2 also tells us that y is PRIME, and we just concluded that the product xy = y
This means the product xy MUST BE PRIME
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

Official Solution:


If \(x\) and \(y\) are integers is \(xy\) a prime number?

Firstly that only positive integers can be prime numbers and that 1 is not a prime number.

Secondly, for the product of two integers \(x\) and \(y\) to be a prime number, either one of them must be \(1\) and the other a prime number, for example (1, 2), (1, 3), (1, 5), (1, 7), ..., or one of them must be \(-1\) and the other \(-\text{prime}\), for example (-1, -2), (-1, -3), (-1, -5), (-1 ,-7), ...

(1) \(x^2=1\).

Either \(x=1\) or \(x=-1\). Not sufficient.

(2) \(y\) is a prime number and \(x\) is a positive number but not a prime number.

If \(y=\text{prime}\) and \(x=1\), then \(xy=y=\text{prime}\). However, if, for example, \(y=\text{prime}\) and \(x=4\), \(xy=4y \neq \text{prime}\). Not sufficient.

(1)+(2) From (1), we know that \(x=1\) or \(x=-1\). From (2), we know that \(x > 0\). Combining these, we get \(x=1\). Therefore, \(xy=1*\text{prime}=\text{prime}\). Sufficient.


Answer: C­

If \(x\) and \(y\) are integers is \(xy\) a prime number?

(1) \(x^2 = 1\)
x = 1 and -1
Nothing about y so many values are possible.

INSUFFICIENT.

(2) \(y\) is a prime number and \(x\) is a positive number but not a prime number.­
A. If x = 1 its a prime
B. If x is a multiple of integers other than 1 then it is not.
 
INSUFFICIENT.

Together 1 and 2.
Case B. of St. 2 works

SUFFICIENT.

Answer C.