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Senior Manager  Status: Do and Die!!
Joined: 15 Sep 2010
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If x and y are positive integers, is x^2*y^2 even ? (1) x +  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 61% (01:30) correct 39% (01:33) wrong based on 225 sessions

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If x and y are positive integers, is x^2*y^2 even ?

(1) x + 5 is a prime number

(2) y + 1 is a prime number

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I'm the Dumbest of All !!

Originally posted by shrive555 on 19 Oct 2010, 10:23.
Last edited by Bunuel on 20 Oct 2010, 13:47, edited 1 time in total.
Edited the stem.
Intern  Joined: 29 Sep 2010
Posts: 13
Schools: IMD '16 (A)
GMAT 1: 730 Q47 V44 Re: Number properties  [#permalink]

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A for me.

First off, an even number squared will give an even number(2^2 = 4, 4^2 = 16 etc). An odd number squared will give an odd number (3^2 = 9, 5^2 = 25 etc). Also, for the multiplication between two numbers to come out even, at least one of the numbers must be even.

We know all primes except for 2 are odd. Since x and y are positive, the only way for x + 5 to be prime will be if x is even (i.e. if x = 1 then x + 5 = 6 which is not prime. But if x = 2 then x + 5 = 7 which is prime). Therefore, x^2 will also be even and x^2y^2 will also be even regardless of what y is. statement 1 is sufficient.

Using a similar approach for condition 2, we can set y = 1 (y + 1 = 2 which is prime) or y = 2 (y + 2 = 3 which is also prime). So we see with this condition, y can be both even or odd. So statement 2 alone is not sufficient.
Math Expert V
Joined: 02 Sep 2009
Posts: 53657
Re: Number properties  [#permalink]

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shrive555 wrote:
If x and y are positive integers, is x2y2 even ?

(1) x + 5 is a prime number

(2) y + 1 is a prime number

In order $$x^2y^2$$ to be even at least one of the unknowns must be even.

(1) $$x+5=prime$$ --> as $$x$$ is a positive integer then this prime can not be the only even prime 2 (in this case $$x+5=2$$ --> $$x=-3=negative$$), so $$x+5=prime=odd$$ --> $$x=odd-5=odd-odd=even$$. Sufficient.

(2) y + 1 is a prime number --> $$y$$ could be 1, so odd, and we won't be sure whether $$x^2y^2=even$$ or $$y$$ could be even (for example 2) and then $$x^2y^2=even$$. Not sufficient.

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Current Student Joined: 15 Jul 2010
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GMAT 1: 750 Q49 V42 Re: Number properties  [#permalink]

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A.

Kind of trivial but I was staring at x2y2 for 20 seconds.

edit it to x^2y^2 for others maybe?
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Current Student D
Joined: 12 Aug 2015
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Schools: Boston U '20 (M)
GRE 1: Q169 V154 Re: If x and y are positive integers, is x^2*y^2 even ? (1) x +  [#permalink]

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Excellent Question
Given info => x,y are positive integers (very important)
We need to check if x^2*y^2 is even or not.
Now x^2*y^2 will be even when either x or y or both are even
Hence we need to find => "If atleast one of x or y is even"
Statement 1
Here the least value of x+5 is 6 (as the least value of x is 1)
Here we need to remember that all the Prime numbers greater than 2 are odd.
Hence x+5 must be odd
so x must be even
BINGO
sufficient

Statement 2

Here y the least value of y+1 is 2

Let us take y = 1 (as y+1=2 which is a prime too)
if x is even => then x^2*y^2 will be even
if x is odd => then x^2*y^2 will be odd
Hence insufficient

Hence A

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Re: If x and y are positive integers, is x^2*y^2 even ? (1) x +  [#permalink]

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shrive555 wrote:
If x and y are positive integers, is x^2*y^2 even ?

(1) x + 5 is a prime number

(2) y + 1 is a prime number

$$x,y\,\, \ge 1\,\,\,{\rm{ints}}\,\,\,\,\left( * \right)$$

$${\left( {xy} \right)^2}\,\,\mathop = \limits^? \,\,\,{\text{even}}\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\boxed{\,\,?\,\,\,:\,\,x\,\,{\text{even}}\,\,\,{\text{or}}\,\,\,y\,\,{\text{even}}\,\,\,\,}$$

$$\left( 1 \right)\,\,\left\{ \matrix{ x + 5\,\,\,\,\mathop \ge \limits^{\left( * \right)} \,\,\,6 \hfill \cr x + 5\,\,{\rm{prime}} \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x + 5\,\, = {\rm{odd}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x\,\,{\rm{even}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle$$

$$\left( 2 \right)\,\,\,y + 1\,\,{\rm{prime}}\,\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Joined: 16 Jul 2018
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Re: If x and y are positive integers, is x^2*y^2 even ? (1) x +  [#permalink]

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shrive555 wrote:
If x and y are positive integers, is x^2*y^2 even ?

(1) x + 5 is a prime number

(2) y + 1 is a prime number

1 is not prime
So, the answer shoud be D as both x and y = 2 Re: If x and y are positive integers, is x^2*y^2 even ? (1) x +   [#permalink] 13 Oct 2018, 13:13
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