Take two arbitrary integers, x and y

We can express the odd integer x as 2a + 1 for some integer a

We can express the odd integer y as 2b + 1 for some integer b

x + y = 2a + 1 + 2b + 1

x + y = 2a + 2b + 2

Factor out a 2:

x + y = 2(a + b + 1)

Since 2 times any integer even or odd is always even, then

We can express the odd integer x as 2a + 1 for some integer a

We can express the odd integer y as 2b + 1 for some integer b

x + y = 2a + 1 + 2b + 1

x + y = 2a + 2b + 2

Factor out a 2:

x + y = 2(a + b + 1)

Since 2 times any integer even or odd is always even, then

**x + y by definition is even**.
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