If x and y are positive integers, is x odd? (1) y/x is a prime number

Question

If x and y are positive integers, is x odd?

(1) y/x is a prime number
(2) x*y is a prime number

M36-18

Bunuel · Accepted Answer

Official Solution:

If  x  and  y  are positive integers, is  x  an odd number? 
 
(1)  \frac{y}{x}  is a prime number
 
This one is clearly insufficient. For example, consider  y=2  and  x=1=odd  or  y=4  and  x=2=even .
 
(2)  x*y  is a prime number.
 
A prime number has only two factors: 1 and itself. So,  x*y=prime  implies that  x=1  and  y=prime  or vise-versa. So,  x  could be odd (when  x=1  and  y=prime ), as well, as even (when  x=2=prime=even  and  y=1 ) Not sufficient.
 
(1)+(2) The case when  y=1  and  x=prime  (from (2)) is not possible because in this case  \frac{y}{x}  would be a proper fraction (a number between 0 and 1), so not a prime as given in (1). Hence we have another case from (2):  x=1=odd  and  y=prime . Sufficient.

Answer: C

SSM700plus · Answer

Let's take statement 1:- y/x is a prime number

4/2 is prime....... y=4 and x=2 (even). Ans is No

6/3 is prime....... y=6 and x =3 (odd). Ans is Yes.

Not sufficient to give definite answer.

Let's take statement 2:- x*y is a prime number

Product of two number is prime only if any one of X & Y is 1.
x =1 and y = 2 ==> x*y = 1 * 2 = Prime. x is odd. Yes.
x =2 and y = 1 ==> x*y = 2 * 1 = Prime. x is even. No.

Hence Not sufficient.

Together Statement 1 and Statement 2

Note that y/x is prime. It also implies that y>x. Hence it is sufficient.

Let's see with examples. Let's use example of statement 2.
x= 1 and y= 2. y/x = 2/1 = 2 is prime. X is odd.
x=2 and y =1. 1/2 = 0. this not prime. So it can't be valid because y/x gives you a ZERO., which is not a prime number.

Hence Both Statement together --> sufficient.

Ans -C.

EgmatQuantExpert · Answer

Solution

Given:  
• x and y are positive integers

To find:  
• Whether the value of x is odd or not

Analysing Statement 1  
• As per the information provided in Statement 1,  \frac{y}{x}  is a prime number 
• This can be feasible with multiple values of x and y
 o If y = 9 and x = 3, then  \frac{y}{x}  = 3, a prime number and x is an odd number 
 o If y = 4 and x = 2, then  \frac{y}{x}  = 2, a prime number and x is an even number 
Hence, from Statement 1 we cannot say whether x is odd or not

Analysing Statement 2  
• As per the information provided in Statement 2, x*y is prime
• If the product of x and y is prime, then one of them has to be a prime number and the other one must be 1
 o Now if we assume x = 1, y can be any prime number 
 o But if we assume y = 1, x can be either even prime or odd prime 
Hence, from Statement 2 we cannot say whether x is odd or not

Combining Both Statements  
• Considering both the statements together, we can say
 o Product of x and y is prime 
 o Ratio of y and x is prime 
• We already know if  x*y  = prime, one of x and y must be prime
 o But if we consider the 2nd statement also, x must be equal to 1 – as any other value of x will violate the 1st condition (also x cannot be equal to y, as  \frac{y}{x}  will become 1, which is not a prime number) 
• As we can say that x is equal to 1, x is an odd number

Hence, the correct answer is option C.

Answer: C

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

GMATin · Answer

We can solve this by using examples. Lets get down to statements.

Statement 1: Easiest numbers to take when a question is testing prime numbers and odd-even concept is to take 2 and 3.

If x=2 and y=6, then y/x=6/2=3 which is prime. Here, x is even

If x=3 and y=6, then y/x=6/3=2, which is prime. Here, x is odd.  Clearly, this statement is insufficient.

Statement 2: For multiplication of two numbers to be prime, one of them has to necessarily be 1 and other has to be a prime number. We are already given x and y are positive integers. So, latter condition is satisfied. For the former, lets take the same cases we have taken before:

If x=2 and y=1, then xy=2 which is prime. Here, x is even

If x=3 and y=1, then xy=3, which is prime. Here, x is odd.  Clearly, this statement is insufficient.

Note here: You could have taken x to be 1 in both cases, but that would really not give you a conclusive proof. Hence, we tested x to be a prime number.

Combining statements 1 & 2: The only way these two statements will be true is if x=1. This is because if x were any other positive integer, y will have to be a multiple of x, in order for statement 1 to give us a prime number. And if that was the case, xy would never be a prime number.

Let me take an example:

If x=2 and y=6, then y/x=6/2=3 which is prime. This satisfies statement 1
But, for statement 2, xy=2*6=12 which is not prime. Hence, statement 2 is not satisfied.

Thus, x has to be 1 which is odd.

Hence, correct answer is C

If x and y are positive integers, is x odd? (1) y/x is a prime number

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If x and y are positive integers, is x odd? (1) y/x is a prime number

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