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Bunuel
If x is a positive odd integer and y is a negative even integer, which of the following must be true?

A. \(x^3 + y\) is a positive odd integer
B. \(x^2 + y^2\) is a negative odd integer
C. \(x^0 + y^{11}\) is a negative odd integer
D. \(x + y\) is a positive odd integer
E. \(x + y\) is a negative odd integer


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Gentle note to all experts and tutors: Please refrain from replying to this question until the Official Answer (OA) is revealed. Let students attempt to solve it first. You are all welcome to contribute posts after the OA is posted. Thank you all for your cooperation!
To simplify calculation let’s assume

positive odd integer x = 1 or 3

Negative even integer y = -2

Must be true should hold for all values.

A. \(x^3 + y\) is a positive odd integer.

case 1: x =1, y=-2 : 1-2 = -1 not a positive odd integer

case 2: x =3, y=-2 3-2 = 1 a positive odd integer.

B. \(x^2 + y^2\) is a negative odd integer

case 1: x =1, y=-2 : (1)^2+ (-2)^2 = 1+4 = 5. Positive odd integer.

case 2: x =3, y=-2 : (3)^2+ (-2)^2 = 9+4 = 13 positive odd integer.

C. \(x^0 + y^{11}\) is a negative odd integer

case 1: x =1, y=-2 : 1^0 + (-2)^11

= (-2)^odd power will result in a negative number greater than 1.

Thus 1 - even number (larger) = Negative odd integer.

case 2: x =3, y=-2 : (3)^0 + (-2)^11 . This is will also yield a negative odd integer.


D. \(x + y\) is a positive odd integer

case 1: x =1, y=-2 : 1-2 = -1 negative odd integer.

case 2: x =3, y=-2 : 3-2 = 1 positive odd integer.

E. \(x + y\) is a negative odd integer

case 1: x =1, y=-2 : 1-2 = -1 negative odd integer.

case 2: x =3, y=-2 : 3-2 = 1 positive odd integer.

Option C is a must be true statement .
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Here what I thought to solve this that in option C, X^0+Y^11; X^0 always +1 for any value and Y^11 gives negative value for all even integers because power is an odd number.
So here difference of +1 from any even negative integers would always give negative Odd integers.
Bunuel
If x is a positive odd integer and y is a negative even integer, which of the following must be true?

A. \(x^3 + y\) is a positive odd integer
B. \(x^2 + y^2\) is a negative odd integer
C. \(x^0 + y^{11}\) is a negative odd integer
D. \(x + y\) is a positive odd integer
E. \(x + y\) is a negative odd integer


­

Gentle note to all experts and tutors: Please refrain from replying to this question until the Official Answer (OA) is revealed. Let students attempt to solve it first. You are all welcome to contribute posts after the OA is posted. Thank you all for your cooperation!
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