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# If x and y are positive integers and xy is divisible by prime number p

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Manager
Joined: 28 Sep 2013
Posts: 85
GMAT 1: 740 Q51 V39
If x and y are positive integers and xy is divisible by prime number p  [#permalink]

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Updated on: 12 Sep 2016, 00:32
2
4
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Difficulty:

65% (hard)

Question Stats:

56% (02:08) correct 44% (01:42) wrong based on 119 sessions

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If x and y are positive integers and xy is divisible by prime number p. Is p an even number?

(1) $$x^2 * y^2$$ is an even number

(2) $$xp = 6$$

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Originally posted by RichaChampion on 12 Sep 2016, 00:15.
Last edited by Bunuel on 12 Sep 2016, 00:32, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
Manager
Joined: 28 Sep 2013
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GMAT 1: 740 Q51 V39
Re: If x and y are positive integers and xy is divisible by prime number p  [#permalink]

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12 Sep 2016, 00:16
This is my reasoning. I believe its E. perhaps Bunuel can tell the OA.

If X and Y are positive integers and X*Y is divisible by prime number P. Is P an even number?

(1) X²∗Y² is an even number
(2) X*P = 6

Statement 1
X²∗Y² is an even number
Inferences →
One among X and Y can be even. That means either X is even or Y is even. In this case, we can't deduce whether P is even or ODD. or→
Both X and Y can be even in this case we must have P, which is a prime number, an even integer = 2.
So from this statement, we get both YES and NO. Thus, this statement is not sufficient.

Statement 1
X*P = 2 X 3 = 3 X 2 = 1 X 6 = 6 X 1
Remeber the one in red is not posisble as 1 is not a prime number. So again here we have YES and NO.

Let us see If by combination we can get anything.

By Combination we know that X and P are opposite. That means If one is odd then other is Even and Vice Versa.
A lot depend on Y now.

If X is Odd then P is Even(=2), but notice that when X is odd Y has to be even in order to maintain XY → Even.
If X is even then P will depend now on Y, but notice here that Y has no constraint now, the constraint is dismissed. Y can be Even or Odd. Thus, P can be even(=2) or odd(any prime number such as 3, 5, 7, 11, 13____).

I think the answer should be E.
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Re: If x and y are positive integers and xy is divisible by prime number p  [#permalink]

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12 Sep 2016, 00:33
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RichaChampion wrote:
If x and y are positive integers and xy is divisible by prime number p. Is p an even number?

(1) $$x^2 * y^2$$ is an even number

(2) $$xp = 6$$

If x and y are positive integers and xy is divisible by prime number p. Is p an even number?

Notice that as given that $$p$$ is a prime number and the only even prime is 2, then the question basically asks whether $$p=2$$.

(1) $$x^2 * y^2$$ is an even number. $$x^2*y^2=\text{even}$$ means that $$xy=\text{even}$$ (this means that at least one of the unknowns is even). We have that some even number is divisible by prime number $$p$$, not sufficient to say whether $$p=2$$, for example if $$xy=6$$ then $$p$$ can be either 2 or 3.

(2) $$xp = 6$$. Since $$x$$ is a positive integer and $$p$$ is a prime number then either $$x=2$$ and $$p=3$$ (answer NO) or $$x=3$$ and $$p=2$$ (answer YES). Not sufficient.

(1)+(2) If $$y=6$$ then $$xy=\text{even}$$, so the first statement is satisfied irrespective of the value of $$x$$ and thus we have no constraints on its value. So from (2) $$x$$ can take any of the two values 2 or 3, which means that $$p$$ can also take any of the two values 2 or 3, respectively. Not sufficient.

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Re: If x and y are positive integers and xy is divisible by prime number p  [#permalink]

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12 Sep 2016, 00:36
Bunuel wrote:
RichaChampion wrote:
If x and y are positive integers and xy is divisible by prime number p. Is p an even number?

(1) $$x^2 * y^2$$ is an even number

(2) $$xp = 6$$

If x and y are positive integers and xy is divisible by prime number p. Is p an even number?

Notice that as given that $$p$$ is a prime number and the only even prime is 2, then the question basically asks whether $$p=2$$.

(1) $$x^2 * y^2$$ is an even number. $$x^2*y^2=\text{even}$$ means that $$xy=\text{even}$$ (this means that at least one of the unknowns is even). We have that some even number is divisible by prime number $$p$$, not sufficient to say whether $$p=2$$, for example if $$xy=6$$ then $$p$$ can be either 2 or 3.

(2) $$xp = 6$$. Since $$x$$ is a positive integer and $$p$$ is a prime number then either $$x=2$$ and $$p=3$$ (answer NO) or $$x=3$$ and $$p=2$$ (answer YES). Not sufficient.

(1)+(2) If $$y=6$$ then $$xy=\text{even}$$, so the first statement is satisfied irrespective of the value of $$x$$ and thus we have no constraints on its value. So from (2) $$x$$ can take any of the two values 2 or 3, which means that $$p$$ can also take any of the two values 2 or 3, respectively. Not sufficient.

Thanks for Confirming.
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Joined: 07 Aug 2016
Posts: 13
Re: If x and y are positive integers and xy is divisible by prime number p  [#permalink]

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12 Sep 2016, 18:45

As if we put X=2 and y=3 or vice Versa both are prime numbers it will give us xy=6 which is divided by both 2 (even) and 3(odd)

X2 * Y2 = (2)2 * (3)2 = 4*9 =36 or vice Versa if X=3 and Y=2 (then also we will get 36) which is even no so satisfying first condition

here the second condition says xp=6 now X can be 2 or 3 as both are prime nos. so if we X=2 then P=3 and if we put X=3 then P=2 . So even by using second condition we are not getting a definite answer whether p is even or odd, so both statements together as well are not sufficient to answer

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Re: If x and y are positive integers and xy is divisible by prime number p  [#permalink]

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09 Aug 2018, 17:50
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Re: If x and y are positive integers and xy is divisible by prime number p &nbs [#permalink] 09 Aug 2018, 17:50
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