GMAT Practice Question: If x and y are integers and x-y is odd, which of the following must be true? I. x y...
Question
If x and y are integers and x-y is odd, which of the following must be true? I. x y is even. II. x^{2}+y^{2} is odd. III. (x+y)^{2} is even.
- I only
- II only
- III only
- I and II only
- I, II, and III
Topics: odds/evens, development formulas, number theory, must be true statements
Solution
Step 1
We first note that x-y is odd only when one of x and y is odd and the other is even. We translate \textbf{\text{"x - y is odd"}} accordingly.
x \text{ is odd and } y \text{ is even} x \text{ is even and } y \text{ is odd}
Step 2
For statement I, since one of x and y is even and the other is odd, their product must be even.
xy = \text{odd} \times \text{even} = \text{even} xy = \text{even} \times \text{odd} = \text{even}
Step 3
For statement II, squaring preserves parity, and then adding an odd and an even yields an odd result.
x^2 = \text{odd} \times \text{odd} = \text{odd} y^2 = \text{even} \times \text{even} = \text{even} x^2 + y^2 = \text{odd} + \text{even} = \text{odd}
Step 4
For statement III, adding an odd and an even gives an odd, and squaring an odd gives an odd, so (x+y)^2 is odd, not even.
x + y = \text{odd} + \text{even} = \text{odd} (x+y)^2 = \text{odd} \times \text{odd} = \text{odd}
Answer
I and II only
