Is x + y odd? (1) 2y + x is even (2) 2x + y is even­

Target question: Is x+y odd?

Statement 1: 2y + x is even
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 2 and y = 1. So, 2y + x = (2)(1) + 2 = 4, which is even. In this case x + y = 2 + 1 = 3, and 3 is ODD
Case b: x = 2 and y = 2. So, 2y + x = (2)(2) + 2 = 6, which is even. In this case x + y = 2 + 2 = 4, and 4 is EVEN
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, you can read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: 2x + y is even
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 1 and y = 2. So, 2x + y = (2)(1) + 2 = 4, which is even. In this case x + y = 1 + 2 = 3, and 3 is ODD
Case b: x = 2 and y = 2. So, 2x + y = (2)(2) + 2 = 6, which is even. In this case x + y = 2 + 2 = 4, and 4 is EVEN
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that 2y + x = even
Statement 2 tells us that 2x + y = even
ADD the equations to get: (2y+x) + (2x+y) = even + even
Simplify: 3x + 3y = EVEN
Factor: 3(x + y) = EVEN
Since we know that (ODD)(EVEN) = EVEN, and since 3 is ODD, we can conclude that x+y must be EVEN
In other words, x+y is NOT odd
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer:

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