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# If x and y are positive integers, is 4x^2 + xy + y^3 even?

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If x and y are positive integers, is 4x^2 + xy + y^3 even?  [#permalink]

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10 Oct 2018, 01:22
1
00:00

Difficulty:

55% (hard)

Question Stats:

57% (01:58) correct 43% (01:58) wrong based on 28 sessions

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If x and y are positive integers, is $$4x^2$$ + xy + $$y^3$$ even?
(1) x – y = 5
(2) y is a prime number greater than 17.
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Re: If x and y are positive integers, is 4x^2 + xy + y^3 even?  [#permalink]

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10 Oct 2018, 01:49
rencsee wrote:
If x and y are positive integers, is $$4x^2$$ + xy + $$y^3$$ even?
(1) x – y = 5
(2) y is a prime number greater than 17.

Question: is $$4x^2$$ + xy + $$y^3$$ even?

$$4x^2$$ is always even as it's multiple of 2 so we need to find whether $$xy + y^3$$ is even

Question REPHRASED: Is $$y *(x + y^2)$$ even?

Statement 1: x – y = 5

i.e. one of x and y must be even and other must be odd
Case 1: If y is even and x is odd, $$y *(x + y^2)$$ will be even
Case 2: If y is odd and x is even, $$y *(x + y^2)$$ will be Odd hence
NOT SUFFICIENT

Statement 2: y is a prime number greater than 17.
i.e. y is Odd but nothing about x is known hence
NOT SUFFICIENT

Combining the two statements
y is odd and x is even, therefore, $$y *(x + y^2)$$ will be Odd hence
SUFFICIENT

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Re: If x and y are positive integers, is 4x^2 + xy + y^3 even?  [#permalink]

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10 Oct 2018, 02:51
rencsee wrote:
If x and y are positive integers, is $$4x^2$$ + xy + $$y^3$$ even?
(1) x – y = 5
(2) y is a prime number greater than 17.

is $$4x^2$$ + xy + $$y^3$$ even

$$4x^2$$ = even........always.

xy = odd / even

$$y^3 = odd / even.$$

As we want even integer , all these terms have to be even or xy and $$y^3$$ both have to be odd.

Statement 1:

x - y = 5

let x be 6 and y be 1

6 - 1 = 5

xy = 6

$$y^3 = 1.$$

Final result will be odd as we have 2 even and a odd integer.

let x be 7 and y be 2.

In this case , we will get and even integer.

So, statement 1 is NOT sufficient ........

Statement 2: y is a prime number greater than 17.

y is a prime greater than 17. it means y is an odd. But we don't know about x. x could be even or odd.

NOT sufficient .

Combining both statements :

Statement 1 states that either x or y has to be odd. 2nd statement states that y is odd. So, x has to be even.

Let x be 60 and y be 23.

The ultimate result is odd.

Sufficient.

Re: If x and y are positive integers, is 4x^2 + xy + y^3 even?   [#permalink] 10 Oct 2018, 02:51
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