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If x and y are positive integers , is the product xy even

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If x and y are positive integers , is the product xy even  [#permalink]

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New post 29 Jan 2009, 05:47
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If x and y are positive integers , is the product xy even

(1) 5x - 4y is even
(2) 6x + 7y is even

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If x and y are positive integers , is the product xy even  [#permalink]

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New post 02 Aug 2016, 13:22
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krishan wrote:
If x and y are positive integers , is the product xy even

(1) \(5x - 4y\) is even
(2) \(6x + 7y\) is even


1) \(4y\) will always be even. Then we have \(5x - even = even\). For this to be the case, \(5x\) must be even. Since 5 can't be even, then x must be even. Thus the product \(xy\) will be even. Sufficient.

2) \(6x\) will always be even. Then we have \(even + 7y = even\). Thus \(7y\) is even, and \(y\) is even, and \(xy\) is even. Sufficient.

Answer: D
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Re: If x and y are positive integers , is the product xy even  [#permalink]

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New post 29 Jan 2009, 06:25
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krishan wrote:
If x and y are positive integers , is the product xy even

(1) 5x - 4y is even
(2) 6x + 7y is even


D. in either case x and y has to be even.

(1) 5x - 4y is even: x has to be even. suff.
(2) 6x + 7y is even: y has to be even. suff.
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Re: If x and y are positive integers , is the product xy even  [#permalink]

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New post 29 Jan 2009, 07:43
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krishan wrote:
If x and y are positive integers , is the product xy even

(1) 5x - 4y is even
(2) 6x + 7y is even


agree with D.

1) 5x - 4y is even -> x must be multiple of 2
5*2k-4y = 2*(any number) even
sufficient
2)6x + 7y is even[/
y should multiple of 2

even
sufficient
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Re: If x and y are positive integers , is the product xy even  [#permalink]

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New post 03 Aug 2016, 10:29
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krishan wrote:
If x and y are positive integers , is the product xy even

(1) 5x - 4y is even
(2) 6x + 7y is even


Target question: Is xy even?

Statement 1: 5x - 4y is even
Let's test all 4 cases
case a: x is even and y is even: In this case 5x-4y is EVEN
case b: x is even and y is odd: In this case 5x-4y is EVEN
case c: x is odd and y is even: In this case 5x-4y is ODD
case d: x is odd and y is odd: In this case 5x-4y is ODD
So, cases a and b are both possible.
In both cases the product xy is even
So, xy must be even
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Aside: If anyone is interested, we have a video (below) on testing possible cases for these question types

Statement 2: 6x+7y is even
Let's test all 4 cases
case a: x is even and y is even: In this case 6x+7y is EVEN
case b: x is even and y is odd: In this case 6x+7y is ODD
case c: x is odd and y is even: In this case 6x+7y is EVEN
case d: x is odd and y is odd: In this case 6x+7y is ODD
So, cases a and c are both possible.
In both cases the product xy is even
So, xy must be even
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer =

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Re: If x and y are positive integers , is the product xy even  [#permalink]

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New post 03 Aug 2016, 11:08
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krishan wrote:
If x and y are positive integers , is the product xy even

(1) 5x - 4y is even
(2) 6x + 7y is even


For xy to be even, either x or y or both must be even.

(1) 5x - 4y is even
E-E=E
O-O=E

Since 4y is even, 5x has to be even. 5 is not even, hence x is even.

xy will be even.

(2) 6x + 7y is even
E+E= E
O+O=E

Since 6x is even, 7y has to be even. And because 7 is not even y has to be even.

xy will be even.

Both statements are sufficient.

D is the answer
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Re: If x and y are positive integers , is the product xy even  [#permalink]

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New post 13 Jul 2017, 00:22
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In my opinion, this type of problem is super easy if you really understand what you're being asked, and then re-frame the question to be more straightforward. It's more about the question than the math.

Given: (x,y) > 0, AND are integers

Original Question: Is xy even?

We know that any number multiplied by an even number is even, so the question is really asking: Do we know if either "X" or "Y" (or both, or neither) is even?

Reframing the question this way is better, in my opinion.



A.) 5x - 4y is even.

For 5x - 4y to be even, both 5x and 4y have to be of the same parity. ("Parity" is the word that describes whether a number is odd or even)
In other words, odd + odd, or even + even, produce even results.



5x - 4y

We know off the bat that "4y" is even, because 4 is even. 4 times anything would be even. So we have ( 5x - EVEN NUMBER = EVEN NUMBER. ) Therefore, we know that "5x" is also even, because 4y (even) and 5x have to be the same parity. (even minus even = even)

In order to make 5x even, "x" has to be even. That's because 5 is odd.

Knowing that "x" is even is enough to answer the question: Do we know if either "X" or "Y" (or both, or neither) is even?

SUFFICIENT.


B.) 6x + 7y is even

Same drill here. For 6x + 7y to be even, both terms have to be the same parity. Since 6 is even, we know that 6x is even. Since 6x is even, we know that 7y is even. Since 7y is even, we know that y is even (since 7 is odd).

Knowing that "y" is even is enough to answer the question: Do we know if either "X" or "Y" (or both, or neither) is even?

SUFFICIENT

The answer is D
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Re: If x and y are positive integers , is the product xy even  [#permalink]

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New post 15 Nov 2017, 03:48
\(S1: 5x - 4y = even => 5x = even+4y => 5x = even => x = even => xy = even. Suff\)

\(S2: 6x+7y = even => 7y = even - 6x =>7y = even => y = even => xy = even. Suff\)

Both S1 and S2 are individually sufficient. Final answer = D
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Re: If x and y are positive integers , is the product xy even  [#permalink]

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New post 24 Dec 2017, 18:09
If x and y are positive integers, is the product xy even
(1) 5x - 4y is even
(2) 6x + 7y is even

1. for 5x-4y = Even, x must be even & y may or may not be even. Since x is even, we can conclude xy is even. SUFFICIENT.
Even – Even = Even
Even – Odd = Odd
Odd – Even = Odd
Odd – Odd = Odd

2. for 6x + 7y is even, y must be even. Since y is even, we can conclude xy is even. SUFFICIENT.
Even + Even = Even
Odd + Even = Odd
Even + Odd = Odd
Re: If x and y are positive integers , is the product xy even &nbs [#permalink] 24 Dec 2017, 18:09
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