Data Sufficiency

VJesus12 wrote: ↑

Sat Jan 01, 2022 12:19 pm

If \(x\) and \(y\) are positive integers, is \(xy\) even?

\((1)\, x^2 + y^2 - 1\) is divisible by \(4.\)
\((2)\, x + y\) is odd.

Answer: D

Source: Official Guide

Some important rules:
#1. ODD +/- ODD = EVEN
#2. ODD +/- EVEN = ODD
#3. EVEN +/- EVEN = EVEN

#4. (ODD)(ODD) = ODD
#5. (ODD)(EVEN) = EVEN
#6. (EVEN)(EVEN) = EVEN

Target question: Is xy even?

Given: x and y are positive integers

Statement 1: x² + y² − 1 is divisible by 4.
In other words, x² + y² − 1 is EVEN
This means x² + y² is ODD.
If x² + y² is ODD, then one of the values (x² or y²) is ODD, and the other value (x² or y²) is EVEN
If one of the values (x² or y²) is ODD, then the individual value (x or y) is ODD.
If the other value (x² or y²) is EVEN, then that other value is EVEN.
So, one value (x or y) is ODD, and the other value is EVEN, which means the product xy is EVEN.
Statement 1 is SUFFICIENT

Statement 2: x + y is odd.
This means one value (x or y) is ODD, and the other value is EVEN, which means the product xy is EVEN.
Statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent