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

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14 May 2018, 01:48
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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  [#permalink]

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Updated on: 14 May 2018, 06:51
Bunuel wrote:
If x and y are positive integers, is x odd?

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

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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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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Re: If x and y are positive integers, is x odd? (1) y/x is a prime number  [#permalink]

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14 May 2018, 05:07
SudhanshuSingh E is the answer. How would C come?
In both statements, x is odd and even. Please explain this.

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Re: If x and y are positive integers, is x odd? (1) y/x is a prime number  [#permalink]

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14 May 2018, 06:09
1

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.

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

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14 May 2018, 06:48
viveknegi wrote:
SudhanshuSingh E is the answer. How would C come?
In both statements, x is odd and even. Please explain this.

Posted from my mobile device

Please read the explanation posted by me and let me know the parts or statement you are unable to understand.

It would be great if you can post your analysis so that I can understand the gap.
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Re: If x and y are positive integers, is x odd? (1) y/x is a prime number  [#permalink]

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14 May 2018, 10:40

s1) y/x is a prime number. i.e 12/6= 2, 9/3=3 so, insufficient.

s2) x*y is prime i.e 1*2=2 , 1*3=3, 3*1=3 or 2*1=2 so , insufficient.

s1 & s2 combined : 2/1 =2 or 3/1 =3 value of x has to be 1 here so answer is C.
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Re: If x and y are positive integers, is x odd? (1) y/x is a prime number  [#permalink]

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15 May 2018, 15:47
According to the explanations above, combining two statements, x has to = 1, but 1 raised to any power is not a prime number. Will the 2 nd stmt be violated?
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Re: If x and y are positive integers, is x odd? (1) y/x is a prime number  [#permalink]

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24 Dec 2018, 02:30
Bunuel wrote:
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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Re: If x and y are positive integers, is x odd? (1) y/x is a prime number  [#permalink]

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29 Dec 2018, 00:49
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.

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Re: If x and y are positive integers, is x odd? (1) y/x is a prime number   [#permalink] 29 Dec 2018, 00:49
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