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# If x and y are integers, and w = x^2y + x + 3y, which of the following

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Joined: 02 Sep 2009
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If x and y are integers, and w = x^2y + x + 3y, which of the following  [#permalink]

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20 Nov 2017, 11:51
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75% (hard)

Question Stats:

42% (01:28) correct 58% (02:31) wrong based on 64 sessions

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If x and y are integers, and $$w = x^2y + x + 3y$$, which of the following statements must be true?

I. If w is even, then x must be even.
II. If x is odd, then w must be odd.
III. If y is odd, then w must be odd.

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II and III

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Math Expert
Joined: 02 Aug 2009
Posts: 6269
If x and y are integers, and w = x^2y + x + 3y, which of the following  [#permalink]

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21 Nov 2017, 08:43
Bunuel wrote:
If x and y are integers, and $$w = x^2y + x + 3y$$, which of the following statements must be true?

I. If w is even, then x must be even.
II. If x is odd, then w must be odd.
III. If y is odd, then w must be odd.

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II and III

Actually I and II can be derived from each other if it were a CR question..
I. If w is even, then x must be even..... MEANS... If x is NOT even, w is NOT even ... MEANS if x is odd, w is odd :- SAME as II
so our answer must have BOTH I and Ii.. ONLY C and E are left

But let's solve ..

Ofcourse many ways to solve but just to understand the logic behind it....

$$w = x^2y + x + 3y$$
there are TWO terms, x and 3y , which contain only one of the two variable AND 1 term containing both variable and that too as a product

1) Therefore, if BOTH the variables x and y are opposite that is one odd and one even
a) the term containing both variables will be EVEN as it will be product of E*O
b) the oher terms will be one E and other O
so SUM = w = E+O+E = O

2) if BOTH the variables x and y are SAME
a) ALL the terms will be EVEN if both are even so SUM= w = E+E+E = E
b) ALL the terms will be ODD if both are odd so SUM = w = O+O+O = O

conclusion
if both x and y are EVEN, w is EVEN or, else, w is ODD

let's see the choices...

I. If w is even, then x must be even.
Refer 2(a), w is even only when both x and y are even....TRUE

II. If x is odd, then w must be odd.
refer 1 and 2(b), when x is odd, w is ODD....TRUE

III. If y is odd, then w must be odd.
Again refer the conclusion or 1 and 2(b), y is ODD means w is ODD....TRUE

E all three true
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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If x and y are integers, and w = x^2y + x + 3y, which of the following  [#permalink]

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21 Nov 2017, 08:45
Bunuel wrote:
If x and y are integers, and $$w = x^2y + x + 3y$$, which of the following statements must be true?

I. If w is even, then x must be even.
II. If x is odd, then w must be odd.
III. If y is odd, then w must be odd.

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II and III

A simple glance at the question will indicate that easy number substitution for $$x$$ & $$y$$ can solve this question

as $$w = x^2y + x + 3y$$. So we can have different scenarios for $$x$$ & $$y$$

1) $$x=0 (even)$$, $$y=1 (odd)$$ ; $$w=3 (odd)$$

2) $$x=0 (even)$$, $$y=0 (even)$$ ; $$w=0 (even)$$

3) $$x=1 (odd)$$, $$y=0 (even)$$ ; $$w=1 (odd)$$

4) $$x=1 (odd)$$, $$y=1 (odd)$$ ; $$w=5 (odd)$$

Now -

I. If $$w$$ is even, then $$x$$ must be even - Always True

II. If $$x$$ is odd, then $$w$$ must be odd. - Always True

III. If $$y$$ is odd, then $$w$$ must be odd. - Always True

Option E
If x and y are integers, and w = x^2y + x + 3y, which of the following &nbs [#permalink] 21 Nov 2017, 08:45
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