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If x and y are positive integers, is x odd? (1) y/x is a prime number 14 May 2018, 01:48
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C
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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

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If x and y are positive integers, is x odd? (1) y/x is a prime number 30 Sep 2021, 03:24
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Bunuel
If x and y are positive integers, is x odd?

(1) y/x is a prime number
(2) x*y is a prime number
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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
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If x and y are positive integers, is x odd? (1) y/x is a prime number Updated on: 14 May 2018, 06:51
Originally posted by SSM700plus on 14 May 2018, 02:22.
Last edited by SSM700plus on 14 May 2018, 06:51, edited 1 time in total.
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Bunuel
If x and y are positive integers, is x odd?

(1) y/x is a prime number
(2) x*y is a prime number
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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.
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If x and y are positive integers, is x odd? (1) y/x is a prime number 14 May 2018, 06:09
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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
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If x and y are positive integers, is x odd? (1) y/x is a prime number 24 Dec 2018, 02:30
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If x and y are positive integers, is x odd?

(1) y/x is a prime number
(2) x*y 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 29 Dec 2018, 00:49
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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
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