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....TRUEII. 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....TRUEE 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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